Use compound-angle formulas for exact values

Ontario Grade 12 Mathematics

Trigonometric Functions

MHF4U study topic B3.2

Some angles do not appear on the usual special-angle list. For example,

75^\circ

is not one of

30^\circ

,

45^\circ

, or

60^\circ

. But it can be written as

45^\circ+30^\circ

. Compound-angle formulas let us use familiar exact values to find exact values for such angles. An exact value is written without rounding, often using fractions and square roots.

What you will learn

Recognize an angle as a sum or difference of familiar angles.
Choose and apply the sine, cosine, or tangent compound-angle formula.
Use exact values from special angles to simplify a trigonometric expression.
Check signs and simplify exact answers correctly.

1. Prerequisite bridge: familiar angles and exact values

A special angle is an angle whose trigonometric ratios can be written exactly using common fractions and square roots. The angles

30^\circ

,

45^\circ

, and

60^\circ

are key examples. Their ratios are useful because they can be combined to evaluate other angles.

You should know these values before using the formulas. For instance,

\sin 30^\circ=\frac12

and

\cos 30^\circ=\frac{\sqrt3}{2}

. At

45^\circ

, both sine and cosine equal

\frac{\sqrt2}{2}

. Tangent is the ratio of sine to cosine, so

\tan 30^\circ=\frac{\sqrt3}{3}

and

\tan 45^\circ=1

.

A compound angle is an angle written as the sum or difference of two angles. The same angle can often be represented in more than one way. Choose a representation whose component angles have known exact values.

\tan\theta=\frac{\sin\theta}{\cos\theta}

Review exact special-angle values before substituting into a formula.
Rewrite the target angle as a sum or difference of familiar angles.
Use degrees consistently, or use radians consistently.

2. The compound-angle formulas

The sine and cosine of a sum are not found by simply adding the two ratios. Instead, use the compound-angle formulas. Notice that the cosine formula has a subtraction between its two products, even when the angle itself is a sum.

For a difference, either use the difference formulas directly or replace the second angle with its negative. The direct forms below make the sign changes clear. The tangent formula applies when its denominator is not zero.

These formulas connect the trigonometric ratios of a combined angle to the ratios of its parts. After choosing the formula, substitute the known exact values and simplify. Do not round intermediate values.

\begin{aligned}\sin(A\pm B)&=\sin A\cos B\pm\cos A\sin B\\\cos(A\pm B)&=\cos A\cos B\mp\sin A\sin B\\\tan(A\pm B)&=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\end{aligned}

For sine of a sum, the middle operation is addition.
For cosine of a sum, the middle operation is subtraction.
For tangent, check that the denominator is nonzero.

3. A visual choice guide

Before calculating, identify the target ratio and a useful decomposition of the angle. The table shows a common choice for each ratio at

75^\circ

. It is a planning aid, not a separate formula: the angle decomposition makes the familiar exact values available.

The table also highlights why choosing the angle parts matters. A decomposition using

45^\circ

and

30^\circ

is useful because their sine, cosine, and tangent values are known exactly.

Match the formula to the ratio being evaluated.
Choose component angles with known exact ratios.

4. Application and checking

Compound-angle formulas are useful when a target angle is not itself a familiar special angle, but can be expressed using familiar ones. They also help find several ratios for the same angle: use the sine formula for sine, the cosine formula for cosine, and the tangent formula for tangent.

Check the quadrant of the target angle when working in degrees. Angles between

0^\circ

and

90^\circ

have positive sine, cosine, and tangent. This is a useful sign check, but it does not replace the calculation.

Keep exact values exact. Simplify square roots and fractions, and do not convert to decimals unless asked. A decimal approximation can help check size, but the exact form is the requested result.

Use the angle’s location to check the signs of your answers.
Do not confuse the plus or minus signs in the cosine formula.
Keep exact answers in fraction and radical form.

Planning a calculation for $75^\circ$

Target	Useful decomposition	Formula to choose
$\sin75^\circ$	$45^\circ+30^\circ$	Sine of a sum
$\cos75^\circ$	$45^\circ+30^\circ$	Cosine of a sum
$\tan75^\circ$	$45^\circ+30^\circ$	Tangent of a sum

Worked example

Find three exact ratios for $75^\circ$

Evaluate

\sin 75^\circ

,

\cos 75^\circ

, and

\tan 75^\circ

exactly.

Choose the angle parts
Write $75^\circ$ as $45^\circ+30^\circ$ . Both component angles have known exact trigonometric values, so this decomposition makes the compound-angle formulas useful.
$75^\circ=45^\circ+30^\circ$
Find the sine
Use the sine-of-a-sum formula. Substitute the special-angle values. The two products are equal, so their sum simplifies to $\frac{\sqrt6+\sqrt2}{4}$ .
$\sin75^\circ=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ=\frac{\sqrt6+\sqrt2}{4}$
Find the cosine
Use the cosine-of-a-sum formula, which subtracts the product of the two sine values. This subtraction gives $\frac{\sqrt6-\sqrt2}{4}$ .
$\cos75^\circ=\cos45^\circ\cos30^\circ-\sin45^\circ\sin30^\circ=\frac{\sqrt6-\sqrt2}{4}$
Find the tangent
Use the tangent-of-a-sum formula. The denominator is not zero, so the formula applies. Simplifying the fraction gives $2+\sqrt3$ .
$\tan75^\circ=\frac{\tan45^\circ+\tan30^\circ}{1-\tan45^\circ\tan30^\circ}=2+\sqrt3$
Check the signs
The angle $75^\circ$ lies between $0^\circ$ and $90^\circ$ , so all three ratios should be positive. The exact answers have the correct signs.

Answer:

\sin75^\circ=\frac{\sqrt6+\sqrt2}{4}

,

\cos75^\circ=\frac{\sqrt6-\sqrt2}{4}

, and

\tan75^\circ=2+\sqrt3

.

Check: The sine and cosine answers are positive. Also, their quotient simplifies to

2+\sqrt3

, matching the tangent answer.

Common mistakes and how to avoid them

Adding the ratios directly, such as treating

\sin(A+B)

as

\sin A+\sin B

.

Correction: Use the full compound-angle formula. It combines products of sine and cosine values.

Using a plus sign between the products in the cosine-of-a-sum formula.

Correction: For a sum, cosine uses subtraction:

\cos(A+B)=\cos A\cos B-\sin A\sin B

.

Rounding familiar-angle values before substituting.

Correction: Keep the exact fractions and square roots throughout the calculation.

Using the tangent formula without checking its denominator.

Correction: The tangent formula is valid only when

1-\tan A\tan B

is nonzero for a sum, or the corresponding difference denominator is nonzero.

Lesson summary

Rewrite the target angle as a sum or difference of familiar angles.
Select the compound-angle formula for the ratio you need.
Substitute exact special-angle values and simplify without rounding.
Check signs and, for tangent, make sure the denominator is nonzero.

Check your understanding

Question 1

Which expression correctly represents

\sin(A+B)

?

$\sin A\cos B+\cos A\sin B$
$\sin A\sin B+\cos A\cos B$
$\sin A\cos B-\cos A\sin B$
correctIndex":0,"explanation":"The sine-of-a-sum formula adds the products $\sin A\cos B$ and $\cos A\sin B$ ."

Show answer and explanation

\sin A\cos B+\cos A\sin B

The sine-of-a-sum formula adds the products

\sin A\cos B

and

\cos A\sin B

.

Question 2

Which familiar-angle sum is a useful way to evaluate

105^\circ

exactly?

$60^\circ+45^\circ$
$80^\circ+25^\circ$
$70^\circ+35^\circ$
correctIndex":0,"explanation":"Both $60^\circ$ and $45^\circ$ have familiar exact trigonometric values, so their sum is a useful decomposition."

Show answer and explanation

60^\circ+45^\circ

Both

60^\circ

and

45^\circ

have familiar exact trigonometric values, so their sum is a useful decomposition.

Question 3

What is the exact value of

\cos60^\circ

?

$\frac12$
$\frac{\sqrt3}{2}$
$\frac{\sqrt2}{2}$
correctIndex":0,"explanation":"The special-angle value is $\cos60^\circ=\frac12$ ."

Show answer and explanation

\frac12

The special-angle value is

\cos60^\circ=\frac12

.

Key terms

Compound angle: An angle written as the sum or difference of two angles.
Exact value: A value written without rounding, often using fractions and square roots.
Special angle: An angle, such as $30^\circ$ , $45^\circ$ , or $60^\circ$ , with familiar exact trigonometric ratios.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation B3.2. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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