Increasing and decreasing: where sine and cosine rise and fall

Explain it like I am five

Picture riding a Ferris wheel that starts at 3 o'clock and turns counterclockwise. Your height is sine and your left-right position is cosine. Increasing means that as the angle grows, the value goes up; decreasing means it goes down.

Follow one lap. From 3 o'clock to 12 (0° to 90°) you rise and drift left: sine increases, cosine decreases. From 12 to 9 (90° to 180°) you fall and keep drifting left: both decrease. From 9 to 6 you fall but drift right: sine decreases, cosine increases. From 6 back to 3 you rise and drift right: both increase.

Example: which is larger, cos 105° or cos 165°? Both angles are in Quadrant II (90° to 180°), where cosine decreases, so the larger angle gives the smaller value: cos 105° > cos 165°, read 'cos 105° is greater than cos 165°'. A calculator in degree mode agrees: cos 105° is about −0.259 and cos 165° about −0.966, and a negative number closer to 0 is larger.

In plain words

Think of climbing a staircase. As you move forward, your height rises. A function is increasing on a stretch when every larger input gives a larger output there. Walking downhill gives the opposite pattern, called decreasing. These words describe change, rather than whether a value is positive or negative. On a circular track, follow the point as its angle grows from 0 to 2π. Sine reads its height, and cosine reads its left or right position. Watch each position separately. Sine rises, falls, falls, then rises in the four quarters. Cosine falls, falls, rises, then rises. The question is how the output moves as you increase the input.

The horizontal graph coordinate is input t; solid sine rises in the first and last quarters, while dashed cosine rises in the second half.

Reminder

Ordering negative numbers. A negative number nearer 0 is larger: − $1 over 4$ > − $3 over 4$ . Picture − $1 over 4$ farther right on a number line.
A common shift preserves order. Subtracting the same number keeps order: 410° < 440° becomes 50° < 80° after subtracting 360° from both.
Periodicity preserves outputs. sin 380° = sin 20° because their inputs differ by 360°. Larger original inputs can therefore have equal outputs.
Input labels. $x sub 1$ and $x sub 2$ mean two different input numbers. For trigonometric functions, we usually call the input t and reserve x for the circle's horizontal coordinate.

The horizontal graph coordinate is input t; solid sine rises in the first and last quarters, while dashed cosine rises in the second half.

Read the output entry in the column under its input or category.

Why it works. Follow each quarter of the circle. From (1, 0) to (0, 1), the point climbs and moves left, so sine increases and cosine decreases. From (0, 1) to (−1, 0), it drops and moves left, so both decrease. From (−1, 0) to (0, −1), it drops but moves right, so sine decreases and cosine increases. From (0, −1) to (1, 0), it climbs and moves right, so both increase. Because height and horizontal position move in those directions throughout each quarter, any two input angles in that quarter give the indicated output order.

RuleOn a chosen interval, increasing means > gives f() > f(); decreasing means > gives f() < f(). On [0, 2π], sin increases on [0, ] and [ , 2π], and decreases on [ , ]. cos decreases on [0, π] and increases on [π, 2π].

The same idea, five ways

Say it

Increasing means a larger input always gives a larger output on the stated stretch. Decreasing means it gives a smaller output.

Write it

Sine and cosine change direction during a full turn, so their increasing and decreasing behavior must be stated on intervals.

In math

$x sub 2$ > $x sub 1$ and increasing: f( $x sub 2$ ) > f( $x sub 1$ )
$x sub 2$ > $x sub 1$ and decreasing: f( $x sub 2$ ) < f( $x sub 1$ )
sin: rises in I and IV, falls in II and III
cos: falls in I and II, rises in III and IV

Moving forward along an uphill or downhill path compares successive heights.

See it

The horizontal graph coordinate is input t; solid sine rises in the first and last quarters, while dashed cosine rises in the second half.

The same idea, other ways

As a staircase

Increasing says that forward steps raise the output; decreasing says they lower it. A staircase can climb while it is still below street level. That is why an increasing output can be negative.

The output rises from −3 to −2 although both outputs are negative.

As a moving point

Trace the track counterclockwise. In III the point goes down and right. Down means decreasing sine, but right means increasing cosine. The same input change has different effects on the two outputs.

Height decreases while horizontal position increases.

As output order

Compare two inputs, not one value in isolation. If 20° < 70° and cosine decreases throughout the stretch containing them, the output order reverses: cos 20° > cos 70°. For an increasing function the input and output orders agree.

20° < 70°

cos decreases in I

cos 20° > cos 70°

Decreasing reverses the order of the two outputs.

t goes from	P moves	cos t	sin t	Result
0 → $π over 2$ (I)	(1, 0) → (0, 1)	1 → 0	0 → 1	sin increasing, cos decreasing
$π over 2$ → π (II)	(0, 1) → (−1, 0)	0 → −1	1 → 0	sin decreasing, cos decreasing
π → $3 π over 2$ (III)	(−1, 0) → (0, −1)	−1 → 0	0 → −1	sin decreasing, cos increasing
$3 π over 2$ → 2π (IV)	(0, −1) → (1, 0)	0 → 1	−1 → 0	sin increasing, cos increasing

.1Sine: follow height

Sine is the moving point's height, like a rider's height on a Ferris wheel. Its graph places input t horizontally and sine vertically. The point can still be below the center while climbing.

Increasing from 0 to $π over 2$ , then again from $3 π over 2$ to 2π.
Decreasing from $π over 2$ to $3 π over 2$ .
Quadrant pattern: increasing, decreasing, decreasing, increasing.

Read sine's height: rise, fall, fall, rise.

Worked exampleTwo heights while climbing

Compare sin 15° and sin 55°. Which height is larger?

Read sine's height: rise, fall, fall, rise.

0° < 15° < 55° < 90°, so both inputs are in I.These are the first quarter's boundaries.
Sine increases in I, so sin 15° < sin 55°.The point rises throughout this quarter.

Answer

sin 55° is larger.

Check The later point is higher on the circle.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Sine increases over the entire first half of a turn.

At the point reaches the top and starts descending.

✓ Instead: Sine increases only up to , then decreases.

Tips and tricks

Trace height with your finger rather than horizontal position.

.2Cosine: follow left or right

Cosine is the moving point's horizontal position, like an east or west address. Moving left makes that address smaller; moving right makes it larger, including when the address is still negative.

Decreasing from 0 to π.
Increasing from π to 2π.
Quadrant pattern: decreasing, decreasing, increasing, increasing.

Cosine decreases throughout the first half and increases throughout the second half.

Worked exampleTwo horizontal positions

Compare cos 205° and cos 255°. Which horizontal coordinate is larger?

Cosine decreases throughout the first half and increases throughout the second half.

180° < 205° < 255° < 270°, so both inputs are in III.Both points lie in the lower left quarter.
Cosine increases in III, so cos 205° < cos 255°.The point moves right from x = −1 toward x = 0.

Answer

cos 255° is larger.

Check Both values are negative, but the later point is farther right and closer to x = 0.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: The point moves downward in III, so cosine decreases.

Downward describes y and therefore sine. Cosine reads x.

✓ Instead: Cosine increases in III because the point moves right.

Tips and tricks

For cosine, think left or right on the circle; on its function graph this becomes down or up.

.3The notes' line y = 3x − 5

A line with rule y = 3x − 5 acts like a running total. Every extra unit of input adds 3 to the output. The subtraction of 5 lowers every total by the same amount and cannot change which total is larger.

y = 3x − 5 is increasing on (−∞, ∞).
If the input increases by a positive amount d, the output increases by 3d, which is positive.

The instructor's line y = 3x − 5 rises as x grows.

Worked exampleComparing sines of two angles beyond two full turns

Without a calculator, decide which is larger: sin( $37 π over 8$ ) or sin( $39 π over 8$ ).

The angle = 832.5° ends at the same point as = 112.5°, two full turns earlier, in quadrant II.

Note that $39 π over 8$ > $37 π over 8$ , so $39 π over 8$ is the larger input. We must decide whether its sine is the larger or the smaller output.A larger angle does not automatically give a larger sine. It depends on whether sine is increasing or decreasing where the angles lie.
Subtract two full turns, 4π = $32 π over 8$ , from both angles: $37 π over 8$ − $32 π over 8$ = $5 π over 8$ and $39 π over 8$ − $32 π over 8$ = $7 π over 8$ .Coterminal angles have the same sine. Both angles are shifted by the same number of turns, so their order is preserved: $7 π over 8$ > $5 π over 8$ .
Locate the reduced angles: $π over 2$ = $4 π over 8$ < $5 π over 8$ < $7 π over 8$ < $8 π over 8$ = π. Both lie in quadrant II, inside [ $π over 2$ , $3 π over 2$ ].A comparison by monotonicity needs both inputs in one interval where sine keeps the same direction of change.
On [ $π over 2$ , $3 π over 2$ ], sine is decreasing.This comes from the motion table. As the point moves from the top of the circle down the left side, its height falls.
Since $7 π over 8$ > $5 π over 8$ and sine is decreasing there, sin( $7 π over 8$ ) < sin( $5 π over 8$ ).Decreasing means the larger input gives the smaller output.
Return to the original angles: sin( $39 π over 8$ ) = sin( $7 π over 8$ ) and sin( $37 π over 8$ ) = sin( $5 π over 8$ ). So sin( $37 π over 8$ ) > sin( $39 π over 8$ ).Each original angle has the same sine as its coterminal reduced angle.

Answer

sin() is larger: sin() > sin().

Check Moving-point check: = 112.5° sits just past the top of the circle, while = 157.5° is much nearer the negative x-axis. The point at 112.5° is higher, so its sine is larger. Both sines are positive, so signs alone could not decide this, and monotonicity was needed. Approximate values agree: sin 112.5° ≈ 0.92 and sin 157.5° ≈ 0.38.

Work to write

$37 π over 8$ − 4π = $5 π over 8$ , $39 π over 8$ − 4π = $7 π over 8$
$π over 2$ < $5 π over 8$ < $7 π over 8$ < π, so both are in quadrant II
sin decreases on [ $π over 2$ , $3 π over 2$ ]
$7 π over 8$ > $5 π over 8$ ⇒ sin( $7 π over 8$ ) < sin( $5 π over 8$ )
sin( $37 π over 8$ ) > sin( $39 π over 8$ )

sin( $37 π over 8$ ) is larger: sin( $37 π over 8$ ) > sin( $39 π over 8$ ).

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: The −5 makes this line decreasing.

Subtracting a fixed amount from every output shifts all outputs equally.

✓ Instead: The positive multiplier 3 makes the output increase as the input increases.

Tips and tricks

Look at how an input change affects the output, rather than at whether the formula contains a minus sign.

.4The notes' square function y = $x^{2}$

Squaring tells you how large a square is when its side has a given distance from 0. On the negative side, moving right brings the input closer to 0 and makes its square smaller. On the positive side, moving right takes it farther from 0 and makes its square larger.

y = $x^{2}$ is decreasing on (−∞, 0].
y = $x^{2}$ is increasing on [0, ∞).
It has no single increasing or decreasing direction over all real inputs.

The notes' square function falls toward input 0 and rises away from 0.

Worked exampleComparing cos() and cos() after removing one full turn

Without a calculator, decide which is larger: cos( $23 π over 9$ ) or cos( $25 π over 9$ ).

The angle = 460° ends on the same terminal side as = 100°, one full turn (360°) less. Both have the same cosine.

Note what is compared: the cosine outputs of two inputs. The input $25 π over 9$ is larger than $23 π over 9$ , but that alone does not tell us which cosine is larger.A larger input can give a larger or a smaller output, depending on whether cosine is increasing or decreasing there.
Remove one full turn, 2π = $18 π over 9$ , from each angle: $23 π over 9$ − $18 π over 9$ = $5 π over 9$ and $25 π over 9$ − $18 π over 9$ = $7 π over 9$ . So cos( $23 π over 9$ ) = cos( $5 π over 9$ ) and cos( $25 π over 9$ ) = cos( $7 π over 9$ ).Cosine repeats every full turn, so subtracting 2π does not change the value.
Compare the reduced inputs: $7 π over 9$ > $5 π over 9$ , so the order is preserved.Both angles were reduced by the same single full turn.
Locate both reduced angles. They satisfy $π over 2$ = $4.5 π over 9$ < $5 π over 9$ < $7 π over 9$ < $9 π over 9$ = π, so both lie in quadrant II, inside [0, π].The comparison by monotonicity requires both inputs to lie in one interval where cosine keeps the same direction of change.
On [0, π] cosine decreases, so the larger input gives the smaller output: $7 π over 9$ > $5 π over 9$ gives cos( $7 π over 9$ ) < cos( $5 π over 9$ ).Decreasing means $x sub 2$ > $x sub 1$ gives f( $x sub 2$ ) < f( $x sub 1$ ).
Return to the original angles: cos( $23 π over 9$ ) = cos( $5 π over 9$ ) > cos( $7 π over 9$ ) = cos( $25 π over 9$ ).The reduced values are equal to the original ones.

Answer

cos() is larger: cos() > cos().

Check Both values are negative because both reduced angles are in quadrant II. As the point moves from (100°) to (140°), it moves farther left on the unit circle, so the x-coordinate, which is the cosine, gets smaller. Approximate values agree: cos 100° ≈ −0.17 and cos 140° ≈ −0.77, and −0.17 > −0.77 because −0.17 is closer to 0.

Work to write

cos( $23 π over 9$ ) = cos( $23 π over 9$ − 2π) = cos( $5 π over 9$ )
cos( $25 π over 9$ ) = cos( $25 π over 9$ − 2π) = cos( $7 π over 9$ )
$π over 2$ < $5 π over 9$ < $7 π over 9$ < π, and cos decreases on [0, π]
$7 π over 9$ > $5 π over 9$ ⇒ cos( $7 π over 9$ ) < cos( $5 π over 9$ )
cos( $23 π over 9$ ) > cos( $25 π over 9$ )

cos( $23 π over 9$ ) is larger: cos( $23 π over 9$ ) > cos( $25 π over 9$ ).

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: All square outputs are positive, so the square function is increasing everywhere.

An output's sign does not tell you its direction of change. Also x = 0 gives output 0.

✓ Instead: The square function decreases on negative inputs and increases on positive inputs.

Tips and tricks

Always state the interval when saying the square function increases or decreases.

Strategy: step by step

1. Read what to compare: a larger angle is the larger input, but its sine or cosine may be a larger or smaller output.
2. Locate both angles. For a comparison by increasing or decreasing, both must lie in one interval where the requested function keeps the same direction of change.
3. Use the quadrant motion table, or combine neighboring quarters with the same direction: cosine decreases through I and II and increases through III and IV.
4. Increasing means the larger input gives the larger output. Decreasing means the larger input gives the smaller output.
5. If both angles are shifted by the same complete turn, their order is preserved. If you reduce them by different numbers of turns, compare the reduced inputs again before using the table. Their original order need not be the reduced order.
6. If the two values have opposite signs, the positive value is larger. If neither one interval nor signs decide the comparison, these rules alone are insufficient; use an additional taught exact-value fact.
7. Check the moving point: higher means larger sine; farther right means larger cosine. For negative outputs, a value closer to 0 is larger, such as − $1 over 4$ > − $3 over 4$ .
Read a reference-table entry in the column under its input or category in the matching picture.

Strategy

Comparing sine or cosine without decimal evaluation

Are the angles outside the first turn?

YesUse a common complete-turn shift if possible. If you reduce separately, compare the reduced order again.

NoUse the displayed inputs.

↓

Are both inputs in one interval where the function only increases or only decreases?

YesLarger input gives larger output for increasing, smaller output for decreasing.

NoCheck the signs of the two outputs.

↓

Does one output have a positive sign and the other a negative sign?

YesThe positive output is larger.

NoUse a taught exact value if available; the direction and sign rules alone may not settle the comparison.

1. Locate both terminal sides and note the requested function.
2. Check whether both inputs lie in one interval with a single direction of change.
3. If so, compare the input order and apply increasing or decreasing.
4. If reducing large inputs, use one common shift where possible. After different shifts, compare the reduced inputs anew.
5. If the interval method does not apply, see whether opposite output signs decide the comparison.

Worked exampleWhich value is larger?

Without a calculator, decide which is larger: (a) cos 20° or cos 70°; (b) sin 200° or sin 250°. The question compares two outputs; it does not require finding their decimal values.

The horizontal graph coordinate is input t; solid sine rises in the first and last quarters, while dashed cosine rises in the second half.

(a) 0° < 20° < 70° < 90°, so both inputs lie in I and 70° is larger.One quarter of the circle covers both angles.
Cosine decreases in I, so cos 20° > cos 70°.As the angle grows, the point moves left and its x-coordinate gets smaller.
(b) 180° < 200° < 250° < 270°, so both inputs lie in III and 250° is larger.Both angles fit between the same two boundaries.
Sine decreases in III, so sin 200° > sin 250°.As the angle grows, the point moves down toward (0, −1), making y smaller.

Answer

(a) cos 20° is larger.
(b) sin 200° is larger.

Check A second check in DEG mode gives cos 20° ≈ 0.940 and cos 70° ≈ 0.342; sin 200° ≈ −0.342 and sin 250° ≈ −0.940, rounded to three places. A negative number nearer 0 is larger: −0.342 > −0.940.

Ladder: from easy to exam-hard. Press Try it first on any rung to hide its steps and use them as hints.

Rung 1Rung 1: a rising positive output

Which is larger, sin 12° or sin 38°? Compare heights in one quadrant.

Height increases through I.

0° < 12° < 38° < 90°.Both inputs lie in I.
Sine increases in I, so sin 12° < sin 38°.A larger input moves the point higher.

Answer

sin 38° is larger.

Check The later point in I has greater height.

Rung 2Rung 2: a falling positive output

Which is larger, cos 22° or cos 68°? Compare horizontal positions in one quadrant.

Horizontal position decreases through I.

0° < 22° < 68° < 90°.Both inputs lie in I.
Cosine decreases in I, so cos 22° > cos 68°.The point moves left as the angle increases.

Answer

cos 22° is larger.

Check The earlier point is farther right.

Rung 3Rung 3: positive and decreasing

Which is larger, sin 115° or sin 165°? The signs match, so use direction of change.

Decreasing sine can remain positive.

90° < 115° < 165° < 180°.Both inputs lie in II.
Sine decreases in II, so sin 115° > sin 165°.The point descends from the top toward the x-axis.

Answer

sin 115° is larger.

Check Both heights are positive, but the later one is lower.

Rung 4Rung 4: negative and decreasing

Which is larger, sin 190° or sin 260°? Compare two negative heights.

A more negative height is smaller.

180° < 190° < 260° < 270°.Both inputs lie in III.
Sine decreases in III, so sin 190° > sin 260°.The later point has moved farther below the center.

Answer

sin 190° is larger.

Check The point at 190° is nearer the x-axis, so its negative height is closer to 0 and larger.

Rung 5Rung 5: an increasing output in IV

Which is larger, cos 290° or cos 340°? Compare horizontal positions while returning to the starting point.

Horizontal position increases through IV.

270° < 290° < 340° < 360°.Both inputs lie in IV.
Cosine increases in IV, so cos 290° < cos 340°.The point moves right toward (1, 0).

Answer

cos 340° is larger.

Check The later point is closer to the far right of the circle.

Rung 6Rung 6: opposite signs settle a cross-quadrant comparison

Which is larger, sin 120° or sin 240°? These inputs occupy different quarters.

Compare above and below when signs differ.

120° lies in II, so sin 120° > 0.The point is above the x-axis.
240° lies in III, so sin 240° < 0.The point is below the x-axis.
sin 120° > sin 240°.Every positive number is larger than every negative number.

Answer

sin 120° is larger.

Check The first height is above the center and the second height is below it.

Rung 7Rung 7: shift both inputs by the same lap

Which is larger, cos 410° or cos 440°? Move both inputs into one common quarter.

A common lap shift lets both inputs use the same decreasing quarter.

410° − 360° = 50° and 440° − 360° = 80°.Subtract the same full turn from both; periodicity preserves the cosine values and this common subtraction preserves order.
0° < 50° < 80° < 90°, so cosine decreases across this whole stretch.Both reduced inputs are in I.
cos 50° > cos 80°, so cos 410° > cos 440°.Use the decreasing order, then replace each reduced value by its equal original value.

Answer

cos 410° is larger.

Check A common shift leaves the original smaller input associated with the reduced smaller input.

Rung 8Rung 8: a larger original input can give an equal output

Compare sin 20° and sin 380°. Does a larger input force a larger output over these different turns?

The two turns have different input sizes and the same sine output.

380° − 360° = 20°.The second angle is one complete turn beyond the first.
sin 380° = sin 20°.Sine repeats every 360°.
The outputs are equal even though 20° < 380°.The increasing property applies on suitable intervals; sine is not increasing across all real inputs.

Answer

sin 20° = sin 380°.

Check Both terminal sides end at the same unit-circle point, so their heights match.

Counterexamples: what it is not, and the tempting wrong moves

✗ Not this: Sine is negative in IV, so it is decreasing there.

Its height rises from −1 toward 0. Rising describes change, even below 0.

✓ Instead: Sine is increasing and negative inside IV.

✗ Not this: Any larger angle has a larger sine.

Sine repeats and changes its direction of change. The increasing definition is used on a stated interval.

✓ Instead: Compare angles inside an interval where sine increases; in a decreasing interval the order reverses.

✗ Not this: Subtract complete turns from each angle independently and keep their original input order.

Different shifts may reverse the reduced order or make the reduced inputs equal. Periodicity preserves values, not the order between independently shifted inputs.

✓ Instead: After independent reductions, compare the new inputs again before using increasing or decreasing.

Tips and tricks

Write both angles in the same strict interval before writing an output inequality.
For cosine, remember left means smaller and right means larger. For sine, down means smaller and up means larger.
On a function graph, the horizontal axis gives the input angle t. It is different from the unit circle's coordinate x = cos t.

Trap. Decreasing does not mean negative. In II, sine decreases while staying positive. In IV, sine increases while staying negative. Also, reducing two angles by different numbers of turns can change their order, so compare their reduced inputs before using a direction rule.

Keep in mind

Decreasing is not the same as negative: in Quadrant I cosine falls from 1 to 0 but stays positive.
Among negative numbers the one closer to 0 is larger: −0.2 > −0.9.
Use the rule only when both angles sit in one quarter where the direction holds: 75° and 105° straddle the top, and in fact sin 75° = sin 105°.

Memory hookSine: up, down, down, up. Cosine: down, down, up, up. Chant it quarter by quarter as the Ferris wheel turns from 3 o'clock.

Flash cards: say the answer out loud, then flip

What does 'f is decreasing on an interval' mean?

A larger input gives a smaller output there: if > , then f() < f().

On [0, 2π], where does sine increase?

On [0, ] and on [ , 2π].

On [0, 2π], where does cosine increase?

On [π, 2π].

Which is larger, sin 25° or sin 55°?

sin 55°: both angles are in Quadrant I, where sine increases.

Which is larger, cos 290° or cos 340°?

cos 340°: both angles are in Quadrant IV, where cosine increases.

Cosine is positive in Quadrant I. Is it increasing there?

No. It falls from 1 to 0, so it is decreasing while positive.

Increasing and decreasing: where sine and cosine rise and fall

.1Sine: follow height

.2Cosine: follow left or right

.3The notes' line y = 3x − 5

.4The notes' square function y = x2x2

Comparing sine or cosine without decimal evaluation

.4The notes' square function y = $x^{2}$