Calculator values: DEG or RAD mode

Explain it like I am five

A kitchen scale must know grams or ounces before its number means anything, and a calculator must know the angle unit. DEG mode reads 62 as 62 degrees. RAD mode reads it as 62 radians, where one radian, the angle whose arc is one radius long, is about 57.3°. So 62 radians is a huge turn, about 3552°.

Find sin 62° to four decimal places. The degree sign means DEG mode; with no sign, this course means radians and RAD mode. Press SIN, 62, close the parenthesis, ENTER: 0.8829475929. The fifth decimal is 4, below 5, so it rounds to 0.8829. The same keys in RAD mode show −0.7391806966, the sine of a different turn.

Many calculators have no csc, sec or cot key, so build them as flips: csc 62° = 1 ÷ sin 62° ≈ 1.1325700507, which rounds to 1.1326. The SIN⁻¹ key above SIN is not cosecant. It runs sine backward: you give it a number and it returns an angle. Round only once, at the end.

In plain words

A kitchen scale must know whether you mean grams or ounces. A calculator likewise needs the angle unit before it can give a trig value. DEG and RAD mode choose degrees or radians. A degree sign, or the word degrees, tells you to use DEG. In this course an angle with no unit written means radians, so use RAD. The same typed number can give different answers in the two modes. Many scientific calculators have sine, cosine and tangent keys. If yours lacks a secant, cosecant or cotangent key, build that function from sine and cosine. Keep all stored digits while calculating. Round the final result to the requested number of decimal places.

DEG keys: sin 30 gives 0.5

RAD keys: sin 30 gives about −0.98803

sec θ =

cot θ =

Choose the unit before pressing a function key; use formulas if a key is missing.

Reminder

Radians to degrees. 10 × $180° over π$ ≈ 572.957795…°, so 10 radians rounds to 572.96°, not 572.97°.
Reciprocal. 1 ÷ $2 over 3$ = $3 over 2$ ; a reciprocal requires a nonzero input.
Decimal places. In 0.7880107536, the fourth decimal is 0 and the fifth is 1, so four-place rounding gives 0.7880.
Undefined fractions. $1 over 0$ is undefined. A large approximate calculator number does not create a defined exact quotient.

Why it works. An angle of 10 radians is 10 × ≈ 572.96°, a different turn from 10°. The calculator uses its mode to interpret the number, so the setting changes the angle it evaluates. Secant is 1 divided by cosine and cosecant is 1 divided by sine. Cotangent is cosine divided by sine, which also works where tangent is undefined but sine is nonzero. Rounding an intermediate value changes the number before later arithmetic, so its error can grow. Retain the stored value and round once, at the end.

RuleUse DEG for degrees and RAD for radians. Here an angle with no unit means radians.
sec θ = requires cos θ ≠ 0; csc θ = and cot θ = require sin θ ≠ 0. Round only the final value, retaining requested trailing zeros.

The same idea, five ways

Say it

Set the angle unit, build a missing function key, then round at the end.

Write it

A calculator approximation is useful only when its angle mode matches the given unit.

In math

52°: DEG; 52: RAD
sec θ = $1 over cos θ$
csc θ = $1 over sin θ$
cot θ = $\frac{\cos θ}{\sin θ}$
Four decimal places: 0.7880107536… ≈ 0.7880

Set a kitchen scale's units before reading a weight, then keep precision until the recipe is complete.

See it

DEG keys: sin 30 gives 0.5

RAD keys: sin 30 gives about −0.98803

sec θ =

cot θ =

Choose the unit before pressing a function key; use formulas if a key is missing.

The same idea, other ways

As a kitchen scale

Typing 24 with the scale set to ounces means something different from 24 grams. Typing 24 into a trig key in DEG mode similarly means a different angle from 24 in RAD mode. The written angle unit determines which setting is right.

sin 24°: DEG, about 0.4067

sin 24: RAD, about −0.9056

The mode changes the meaning of the input, even though the typed digits match.

As matching distances on a circle

30 radians is 30 × $180° over π$ ≈ 1718.87°, about 4.8 full turns. It is not the 30° slice. That is why the table's keys sin 30 give 0.5 in DEG but about −0.98803 in RAD. Except for the exact value 0.5, its listed outputs are rounded to five decimal places.

Thirty radians includes several full turns and ends in Quadrant IV.

As a recipe with ingredients

You can build secant from cosine, cosecant from sine, and cotangent from both. Evaluate the needed ingredient without rounding, then do the division. For sec 67°, use 1 ÷ cos 67°; rounding the cosine to 0.39 first changes the final four-place answer.

cos 67° ≈ 0.3907311285

1 ÷ cos 67° ≈ 2.5593046652

1 ÷ 0.39 ≈ 2.5641025641

Keep stored precision.

A rounded ingredient can produce a different final answer.

Keys typed	DEG mode	RAD mode
sin 30	0.5	−0.98803
sin 15	0.25882	0.65029
cos 30	0.86603	0.15425

.1Choose degrees or radians

Before weighing something, set the kitchen scale's unit. Before evaluating a trig function, set the calculator's angle unit. Degrees may appear as ° or as the word degrees. Radians may be written explicitly, but their unit is often omitted. In this course, a bare real number is radians even when it looks like a familiar degree angle.

A degree sign or stated degree unit means DEG.
Radian inputs, including a bare number or a fraction containing π with no degree unit, mean RAD.
The table's decimal outputs are rounded to five places except the exact 0.5.

24° means DEG

24 means RAD

means RAD

Choose mode from the unit, not the size or appearance of the number.

Worked exampleSame digits, different angles

Find sin 24° and sin 24 to four decimal places. In words, evaluate one degree angle and one radian angle.

DEG: sin 24° ≈ 0.4067

RAD: sin 24 ≈ −0.9056

A changed mode produces a changed angle, not a calculator mistake.

For sin 24°, set DEG: the stored value is about 0.4067366431.The degree sign identifies the angle unit.
Round to 0.4067.The fifth decimal is 3, so the fourth stays 7.
For sin 24, set RAD: the stored value is about −0.9055783620.The missing unit means radians here.
Round the magnitude to 0.9056, then restore the minus sign.The fifth decimal is 7, which rounds the fourth from 5 to 6.

Answer

sin 24° ≈ 0.4067
sin 24 ≈ −0.9056

Check The radian input is about 1375.10° and ends in IV, where sine is negative.

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

✗ Not this: A π symbol always means a calculator can choose the mode itself.

Most calculators still interpret the input using their selected angle mode.

✓ Instead: Set RAD explicitly for an input such as −.

Tips and tricks

Recheck mode whenever the next problem switches units.

.2Build a missing secant, cosecant or cotangent key

If a machine has no button for the exact drink you want, combine the ingredients you can get. Many calculators supply sine and cosine, and those are enough to build the remaining functions. Divide 1 by cosine for secant, 1 by sine for cosecant, or cosine by sine for cotangent. Check the denominator before dividing.

sec θ = $1 over cos θ$ when cos θ ≠ 0.
csc θ = $1 over sin θ$ and cot θ = $\frac{\cos θ}{\sin θ}$ when sin θ ≠ 0.
cot θ = $1 over tan θ$ may be used only when tangent exists and is nonzero.

sec: 1 ÷ cosine

csc: 1 ÷ sine

cot: cosine ÷ sine

Sine and cosine are enough to build all three functions.

Worked exampleBuild cosecant and an axis cotangent

Find csc 38° to four decimals and cot 450° exactly. In words, use sine and cosine instead of relying on special keys.

At this axis point cotangent exists while tangent does not.

Set DEG and evaluate sin 38°, keeping its stored digits.Both inputs are degree angles.
csc 38° = 1 ÷ sin 38° ≈ 1.6242692455, so 1.6243.Sine is nonzero, so the reciprocal exists; the fifth decimal is 6.
450° ends at (0, 1), so cot 450° = cos 450° ÷ sin 450° = 0 ÷ 1 = 0.Cotangent's coordinate formula works here even though tangent is undefined.

Answer

csc 38° ≈ 1.6243
cot 450° = 0

Check The cosecant is greater than 1; multiplying its unrounded value by sin 38° gives 1. At 450°, y = 1 is a valid cotangent denominator.

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

✗ Not this: cot 450° is undefined because tan 450° is undefined.

Cotangent is , and this point has x = 0, y = 1.

✓ Instead: cot 450° = = 0.

Tips and tricks

cos ÷ sin is the safest cotangent formula across all its defined inputs.

.3Round after the final operation

A rounded ingredient can change the recipe's result. Keep the calculator's stored digits while taking a reciprocal or multiplying values. After the last operation, count the requested decimal places and inspect one more digit. For a negative answer, round its positive size by the same rule, then restore the minus sign. Keep trailing zeros when the question requests that many places.

For p decimal places, inspect decimal digit p + 1.
Digits 0 through 4 leave the last retained digit unchanged; digits 5 through 9 round the magnitude up, carrying if needed.
Keep requested trailing zeros: 0.7880 shows four decimal places.

0.7880107536…

Keep four places: 0.7880

Next digit: 1, so keep the 0

Final answer: 0.7880

The fifth decimal decides what happens to the fourth.

Worked exampleDelay the reciprocal rounding

Find sec 73° to four decimals. In words, take a reciprocal before rounding it.

sec 73° ≈ 3.4203036198

Four places: 3.4203

Fifth digit: 0, so do not increase

The reciprocal is formed before the final rounding.

Set DEG and keep the full stored value of cos 73°.The degree sign chooses DEG, and premature rounding changes the divisor.
Compute 1 ÷ cos 73° ≈ 3.4203036198.Secant is the reciprocal of the nonzero cosine.
Keep 3.4203; the fifth decimal is 0, so the fourth stays 3.Four-place rounding uses the next digit only after the last calculation.

Answer

sec 73° ≈ 3.4203

Check Using the unrounded values, sec 73° × cos 73° = 1; the rounded answer is an approximation.

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

✗ Not this: The four-place answer to sin 52° is 0.788.

Its value matches 0.7880, but its displayed precision has only three decimal places.

✓ Instead: Write 0.7880 when four decimal places are requested.

Tips and tricks

Use the calculator's stored answer rather than retyping a rounded display.

Strategy: step by step

Read the angle unit from the degree sign or words. If none is written, this course means radians.
Set DEG or RAD before entering the angle, and use parentheses around an entire fraction, negative angle or multiple of π.
Use the function key if available. Otherwise compute sec = 1 ÷ cos, csc = 1 ÷ sin, cot = cos ÷ sin, all with the same angle and mode.
Check whether an exact axis input makes the needed denominator zero. If it does, write undefined rather than trusting a very large calculator display.
Keep the unrounded value in calculator memory while doing reciprocals or other arithmetic.
For p decimal places, inspect digit p + 1: 0 through 4 leaves digit p unchanged; 5 through 9 rounds the magnitude up. Restore the sign and keep exactly p places.

Strategy

Compute and round a trig value

Is the angle given in degrees?

YesSet DEG.

NoSet RAD for radians; an unwritten unit here means radians.

↓

Does the required function divide by zero at an exact axis angle?

YesWrite undefined using the point formulas.

NoEvaluate the function in the chosen mode.

↓

Is a needed sec, csc or cot key absent?

YesUse , or , with the same input and mode.

NoUse the key and check the result against the definition.

↓

Are there later arithmetic steps?

YesKeep the stored result unrounded until they are complete.

NoRound to the requested decimal places and retain trailing zeros.

Read the angle unit from the degree sign or words. If none is written, this course means radians.
Set DEG or RAD before entering the angle, and use parentheses around an entire fraction, negative angle or multiple of π.
Use the function key if available. Otherwise compute sec = 1 ÷ cos, csc = 1 ÷ sin, cot = cos ÷ sin, all with the same angle and mode.
Check whether an exact axis input makes the needed denominator zero. If it does, write undefined rather than trusting a very large calculator display.
Keep the unrounded value in calculator memory while doing reciprocals or other arithmetic.
For p decimal places, inspect digit p + 1: 0 through 4 leaves digit p unchanged; 5 through 9 rounds the magnitude up. Restore the sign and keep exactly p places.

Worked exampleFive original calculator approximations

Round to four decimal places: (a) sin 52°, (b) sin 52, (c) cos(− $7 π over 6$ ), (d) tan 1.3, (e) sec 67°. In words, identify the angle unit before computing each decimal.

52°: DEG

52: RAD

sec 67° = 1 ÷ cos 67°

Round the final value.

The degree sign determines the mode before arithmetic begins.

52°: DEG

52: RAD

sec 67° = 1 ÷ cos 67°

Round the final value.

The degree sign determines the mode before arithmetic begins.

Reference table: read the DEG mode entry directly under its keys typed label.

Reference table: read the RAD mode entry directly under its keys typed label.

(a) DEG mode: sin 52° ≈ 0.7880107536, so 0.7880.The degree sign selects DEG; the fifth decimal is 1.
(b) RAD mode: sin 52 ≈ 0.9866275920, so 0.9866.No unit means radians; the fifth decimal is 2.
(c) RAD mode: cos(− $7 π over 6$ ) ≈ −0.8660254038, so −0.8660.Enter the entire angle in parentheses; the fifth decimal is 2.
(d) RAD mode: tan 1.3 ≈ 3.6021024480, so 3.6021.The fifth decimal is 0; preserve four digits after the decimal point.
(e) DEG mode: sec 67° = 1 ÷ cos 67° ≈ 2.5593046652, so 2.5593.If there is no secant key, use 1 ÷ cos 67°. Keep cosine's stored precision until the final rounding.

Answer

(a) 0.7880
(b) 0.9866
(c) −0.8660
(d) 3.6021
(e) 2.5593

Check The sine and cosine outputs lie between −1 and 1. Secant has size at least 1, and multiplying the unrounded result in (e) by cos 67° returns 1.

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: one degree input

Find sin 18° to four decimal places. In words, choose the degree mode and round one output.

An acute first-quadrant input has a positive sine.

Set DEG and evaluate sin 18° ≈ 0.3090169944.The degree sign fixes the unit.
Keep four places, 0.3090; the fifth decimal is 1.A fifth digit below 5 leaves the fourth unchanged.

Answer

sin 18° ≈ 0.3090

Check The result lies between 0 and 1 because 18° is acute.

Rung 2Rung 2: a negative radian angle with π

Find cos(− $7 π over 6$ ) to four decimals. In words, enter the whole negative radian angle and round its cosine.

The negative rotation ends in Quadrant II, where cosine is negative.

Set RAD and enter cos(−7π ÷ 6), with the entire angle inside parentheses.The input is radians and the minus sign belongs to the whole angle.
The result is approximately −0.8660254038, which rounds to −0.8660.The fifth decimal is 2; retain four places and the minus sign.

Answer

cos(−) ≈ −0.8660

Check − is −210°, coterminal with 150°. Since 150° = 180° − 30°, reflect the 30° point across the x-axis, then take a half turn. Reflection keeps its cosine ; the half turn negates it to −. This agrees with the negative decimal.

Rung 3Rung 3: cotangent without a cotangent key

Find cot 1.1 to four decimals. In words, divide cosine by sine of the same radian input.

The acute first-quadrant angle gives a positive cotangent.

Set RAD, then evaluate cos 1.1 and sin 1.1 without rounding them.No angle unit means radians; both functions must use the same input and mode.
Divide cos 1.1 by sin 1.1 to obtain approximately 0.5089681052.Cotangent equals cosine divided by sine, and sine is nonzero here.
Round to 0.5090.The fifth decimal is 6, increasing 0.5089 to 0.5090 with a carry.

Answer

cot 1.1 ≈ 0.5090

Check 1.1 radians is about 63.03°, in I; both coordinates and cotangent are positive.

Rung 4Rung 4: two modes and a reciprocal

Find sin 52°, sin 52 and sec 67° to four decimals. In words, switch modes deliberately and form the final reciprocal before rounding.

52°: DEG becomes 0.7880

52: RAD becomes 0.9866

67°: DEG, reciprocal becomes 2.5593

The mode can change between neighboring parts of one question.

Set DEG: sin 52° ≈ 0.7880107536, so write 0.7880.The degree sign chooses DEG; the fifth decimal is 1.
Switch to RAD: sin 52 ≈ 0.9866275920, so write 0.9866.The bare input is radians; the fifth decimal is 2.
Switch to DEG and compute 1 ÷ cos 67° using the stored cosine: approximately 2.5593046652.Secant uses the reciprocal of cosine and the angle is degrees.
Round sec 67° to 2.5593.The fifth decimal is 0, so the fourth stays 3.

Answer

sin 52° ≈ 0.7880
sin 52 ≈ 0.9866
sec 67° ≈ 2.5593

Check The sines lie between −1 and 1. The secant exceeds 1, and its unrounded product with cos 67° is 1.

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

✗ Not this: Use DEG for sin 52 because 52 is a whole number.

A whole-number input can be radians. No written unit means radians in this course.

✓ Instead: sin 52° ≈ 0.7880 in DEG; sin 52 ≈ 0.9866 in RAD.

✗ Not this: Always calculate cot θ as 1 ÷ tan θ.

At 450°, tangent is undefined but cotangent is 0. The reciprocal-of-tangent expression cannot be used there.

✓ Instead: Use cot θ = cos θ ÷ sin θ when sine is nonzero. At 450°, it is 0 ÷ 1 = 0.

✗ Not this: Round cos 67° to 0.39 before finding sec 67°.

1 ÷ 0.39 ≈ 2.5641 differs from the correct four-place answer.

✓ Instead: Use the unrounded cosine to obtain sec 67° ≈ 2.5593.

Tips and tricks

For each question write DEG or RAD before entering a number.
Parenthesize the entire angle: cos(−7π ÷ 6) keeps the negative and division together.
Sine and cosine must lie between −1 and 1. A defined secant or cosecant has magnitude at least 1.
0.7880 has four decimal places. Its trailing zero records the requested precision.
Some devices return tiny errors for exact axis values, such as cosine near zero instead of zero. Use the axis point to decide zero and undefined values.
Read reference tables by columns: keep the input label and its output in the same vertical column.

Trap. Leaving the calculator in the wrong mode. Before any calculator question, look for a degree sign and set the mode. Quick test: sin 30 should show 0.5 in DEG mode.

Keep in mind

Test the mode first: sin 90 shows 1 in DEG mode but about 0.8940 in RAD mode.
SIN⁻¹ is not csc: SIN⁻¹ takes a number such as 0.5 and returns an angle, 30° in DEG mode, while csc 30° = 1 ÷ 0.5 = 2.
Round once, at the end: rounding sin 62° to 0.88 first gives 1 ÷ 0.88 ≈ 1.1364 instead of the correct 1.1326.
At an exact axis angle, trust the axis point, not the screen: tan(− $π over 2$ ) in RAD mode can show a huge number, but − $π over 2$ points straight down, where x = 0, so it is undefined.

Memory hookDegree sign? DEG. No sign? RAD. No csc, sec or cot key? Flip: csc = 1 ÷ sin, sec = 1 ÷ cos, cot = cos ÷ sin.

Flash cards: say the answer out loud, then flip

What do DEG mode and RAD mode do?

DEG reads the typed angle as degrees. RAD reads it as radians.

An angle is written with no degree sign. Which mode?

RAD. In this course no unit means radians.

Find sin 62 (no degree sign) to four decimal places.

RAD mode: sin 62 ≈ −0.7392

Find cot 33° to four decimal places.

DEG mode: cos 33° ÷ sin 33° ≈ 1.5399

Is the SIN⁻¹ key the same as csc?

No. SIN⁻¹ runs sine backward and returns an angle. csc θ = 1 ÷ sin θ.

You want sin 90°, but the screen shows 0.8940. What went wrong?

The calculator is in RAD mode. In DEG mode sin 90° = 1.