Teacher Guide: Sum and Difference Identities

Learn the sum and difference formulas for sine, cosine and tangent, find exact values like sin 75° and combine angles from different quadrants.

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For Teachers

Learning Objectives

State the sum and difference identities for sine, cosine and tangent
Find exact values of angles like 15°, 75°, 105° and 165° from special angles
Find sin(A + B), cos(A + B) and tan(A + B) from given values, using quadrant signs
Recognize an expanded expression such as sin 70° cos 20° + cos 70° sin 20° and simplify it
Derive the double angle identities from the sum identities

Prerequisites

• Exact values of sin, cos and tan for 30°, 45°, 60° and the unit circle
• Pythagorean identities: $\sin^2\theta + \cos^2\theta = 1$
• Signs of trigonometric functions in the four quadrants
• Cofunction identities and even and odd identities
• Simplifying fractions with square roots

Discussion Starters

1. Why can sin 30° + sin 60° not be the sine of any angle?
2. Why does the cosine formula flip the sign while the sine formula keeps it? Test A = B = 45°.
3. 75° = 45° + 30° and 75° = 120° - 45°. Will both give the same exact value? Why?
4. How can you get the double angle formulas from the sum formulas in one step?
5. In which real situations do two angles add up, and where would you need these formulas?

Common Misconceptions

Trig functions distribute over sums: $\sin(A + B) = \sin A + \sin B$

Remediation: Test with $A = 30^\circ$ , $B = 60^\circ$ : the left side is 1, the right side about 1.366. Then use the graph with sliders: the red curve $\sin x + \sin b$ separates from $\sin(x + b)$ for every $b$ except $0^\circ$ .

The signs in all formulas match the sign between A and B

Remediation: Have students test $\cos(45^\circ + 45^\circ) = \cos 90^\circ = 0$ with both sign choices. Only the minus version gives 0. Summarize: sine keeps, cosine flips, tangent keeps on top and flips below.

A value found with $\sqrt{1 - \sin^2 A}$ is always positive

Remediation: Draw the angle in its quadrant on the unit circle before calculating, and write the sign first. Compare the final answer with a calculator using the actual angle.

Differentiation Ideas

For Struggling Students:

• Give a formula card and a table of special values, focus on sine and cosine of sums first
• Practise only exact values (75°, 15°) before quadrant problems
• Check every exact result with a calculator to build confidence

For On-Level Students:

• Find all exact values of 15°, 75°, 105°, 165° for sine, cosine and tangent
• Solve quadrant problems for sin(A ± B) and cos(A ± B)
• Simplify expressions by recognizing an expanded identity

For Advanced Students:

• Prove the cosine difference identity with the distance formula on the unit circle
• Derive the triple angle formula sin 3A = 3 sin A - 4 sin³A from sin(2A + A)
• Write a cos t + b sin t as one wave R sin(t + φ) and find R and φ

Standards Alignment

F-TF.C.9 (CCSS.MATH.CONTENT.HSF.TF.C.9)
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems
F-TF.A.3 (CCSS.MATH.CONTENT.HSF.TF.A.3)
Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6

Lesson Resources

visualSum Identity Graph
Sliders for a and b show that sin(a + b) equals sin a cos b + cos a sin b, while sin a + sin b does not
visualStep by Step Proofs
Derive sin(x + 90°) = cos x, cos(180° - x) = -cos x and tan(x + 45°) one rule at a time, with a numeric check
activityExact Value Hunt
Groups build 15°, 75°, 105°, 165° from special angles in two different ways and compare results
worksheetQuadrant Combinations
Given sin A and cos B with quadrants, find sin(A ± B), cos(A ± B) and the quadrant of A + B

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

The sum and difference identities give the sine, cosine and tangent of

A + B

A - B

from the values of

A

and

B

alone.

Sine

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

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

Cosine

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

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

Tangent

\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}

How to remember the signs:

Sine keeps the sign: $+$ in $A + B$ gives $+$ in the formula, and it mixes sine with cosine.
Cosine flips the sign: $A + B$ gives a minus, $A - B$ gives a plus, and it pairs cosine with cosine, sine with sine.
Tangent: the numerator keeps the sign, the denominator flips it.

Important:

\sin(A + B)

is not

\sin A + \sin B

. For example

\sin(30^\circ + 60^\circ) = \sin 90^\circ = 1

, but

\sin 30^\circ + \sin 60^\circ \approx 1.366

With the special angles

30^\circ

45^\circ

and

60^\circ

these formulas give exact values for new angles such as

15^\circ = 45^\circ - 30^\circ

75^\circ = 45^\circ + 30^\circ

105^\circ = 60^\circ + 45^\circ

and

165^\circ = 120^\circ + 45^\circ

. If you set

B = A

, you get the double angle identities.

Worked Examples

Find the exact value of

\sin 75^\circ

Write $75^\circ$ as a sum of special angles

75^\circ = 45^\circ + 30^\circ

→

A = 45^\circ

B = 30^\circ

Apply the sine sum identity

\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ

→ Two products of known values

Substitute the exact values

\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}

→

\frac{\sqrt{6} + \sqrt{2}}{4}

Check with a calculator

\frac{\sqrt{6} + \sqrt{2}}{4} \approx \frac{2.4495 + 1.4142}{4} \approx 0.9659

and

\sin 75^\circ \approx 0.9659

→ The values agree

Answer: $\sin 75^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$

Common Mistakes

Writing $\sin(A + B) = \sin A + \sin B$

Why it's wrong: Sine is not a linear function, so you cannot split it over a sum. With $A = 30^\circ$ and $B = 60^\circ$ : $\sin 90^\circ = 1$ , but $\sin 30^\circ + \sin 60^\circ \approx 1.366$ , which is not even a possible sine value.

Correct: $\sin(A + B) = \sin A \cos B + \cos A \sin B$ . Move the sliders in the graph: the red curve $\sin x + \sin b$ leaves the other two.

Keeping the sign in the cosine formula: $\cos(A + B) = \cos A \cos B + \sin A \sin B$

Why it's wrong: Cosine flips the sign. Test it with $A = B = 45^\circ$ : $\cos 90^\circ = 0$ , but $\cos^2 45^\circ + \sin^2 45^\circ = 1$ .

Correct: $\cos(A + B) = \cos A \cos B - \sin A \sin B$ and $\cos(A - B) = \cos A \cos B + \sin A \sin B$ .

Taking every missing value as positive, e.g. $\cos A = \frac{4}{5}$ when $\sin A = \frac{3}{5}$ and $A$ is in Quadrant II

Why it's wrong: The Pythagorean identity only gives the size of the value. The quadrant decides the sign, and a wrong sign changes the whole result.

Correct: Use the quadrant: in Quadrant II cosine is negative, so $\cos A = -\frac{4}{5}$ . Remember: Quadrant I all positive, II only sine, III only tangent, IV only cosine.

Using $1 + \tan A \tan B$ in the denominator of $\tan(A + B)$

Why it's wrong: In the tangent formula the numerator keeps the sign but the denominator flips it, just like in the cosine formula.

Correct: $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ and $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ .

Why It Matters

The sum and difference identities are the parent formulas of most other trigonometric identities:

Exact values: a calculator gives $\sin 75^\circ \approx 0.9659$ , but the identity gives the exact value $\frac{\sqrt{6} + \sqrt{2}}{4}$ , which is what exams and proofs need.
More identities: double angle, half angle and the cofunction rules all follow from them (set $B = A$ , or $A = 90^\circ$ ).
Waves and sound: physicists rewrite $3\sin t + 4\cos t$ as one wave $5\sin(t + \varphi)$ to read off its amplitude.
Rotations: every computer game and robot arm turns points with $x' = x\cos\beta - y\sin\beta$ and $y' = x\sin\beta + y\cos\beta$ , which is exactly the sum identity.
Angles between lines: the tangent difference identity gives the angle between two roads, ramps or roof lines from their slopes.

Real World Applications

Combining Two Waves

In physics and electronics, a sum like $3\sin t + 4\cos t$ describes two overlapping waves. The sine sum identity shows that it is one single wave $R\sin(t + \varphi)$, and $R$ is its amplitude (how loud or how strong it is).

Example:

$R\sin(t + \varphi) = R\cos\varphi \sin t + R\sin\varphi \cos t$ . Comparing with $3\sin t + 4\cos t$ gives $R\cos\varphi = 3$ and $R\sin\varphi = 4$ .

1Try It Yourself

A signal is $3\sin t + 4\cos t$ . You want to write it as $R\sin(t + \varphi)$ with $R > 0$ .

What is the amplitude $R$ ?

Step 1: Write the mathematical expression

Use $R^2 = (R\cos\varphi)^2 + (R\sin\varphi)^2$ :

Rotating Points in Games and Robotics

A point at distance $r$ and angle $\alpha$ has coordinates $x = r\cos\alpha$, $y = r\sin\alpha$. Turning it by $\beta$ gives the angle $\alpha + \beta$, and the sum identities turn that into the rotation formulas $x' = x\cos\beta - y\sin\beta$ and $y' = x\sin\beta + y\cos\beta$.

Example:

Rotating $(1, 0)$ by $90^\circ$ : $x' = 1 \cdot 0 - 0 \cdot 1 = 0$ and $y' = 1 \cdot 1 + 0 \cdot 0 = 1$ , so the point moves to $(0, 1)$ .

2Try It Yourself

A robot arm holds a tool at $(4, 3)$ . It turns by an angle $\beta$ with $\cos\beta = 0.8$ and $\sin\beta = 0.6$ .

What is the new $y$ -coordinate of the tool?

Step 1: Write the mathematical expression

Use $y' = x\sin\beta + y\cos\beta$ :

The Angle Between Two Roads

A line with slope $m$ makes an angle $\theta$ with the $x$-axis where $\tan\theta = m$. The tangent difference identity gives the angle between two lines straight from their slopes, which planners use for road junctions, ramps and roof lines.

Example:

Lines with slopes $1$ and $0$ : $\tan(\alpha - \beta) = \frac{1 - 0}{1 + 0} = 1$ , so they meet at $45^\circ$ .

3Try It Yourself

On a map, one road has slope $3$ and a second road has slope $\frac{1}{2}$ .

What is the angle between the roads, in degrees?

Step 1: Write the mathematical expression

Use $\tan(\alpha - \beta) = \frac{\tan\alpha - \tan\beta}{1 + \tan\alpha \tan\beta}$ with $\tan\alpha = 3$ , $\tan\beta = \frac{1}{2}$ :

Key Takeaways

1 $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ : sine keeps the sign and mixes sine with cosine
2 $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$ : cosine flips the sign
3 $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$ : the numerator keeps the sign, the denominator flips it
4Exact values of $15^\circ$ , $75^\circ$ , $105^\circ$ , $165^\circ$ come from sums and differences of $30^\circ$ , $45^\circ$ , $60^\circ$ (and $120^\circ$ , $135^\circ$ )
5With angles in other quadrants, find the missing sine or cosine with the Pythagorean identity and take the sign from the quadrant
6 $\sin(A + B) \neq \sin A + \sin B$ ; setting $B = A$ gives the double angle identities

Frequently Asked Questions

Why is there no simple rule like $\sin(A + B) = \sin A + \sin B$ ?

Because sine is not a linear function. Doubling an angle does not double its sine:

\sin 30^\circ = 0.5

, but

\sin 60^\circ \approx 0.866

, not

1

. The sum identity is the correct replacement for that wrong rule.

How are the sum identities connected to the double angle identities?

Set

B = A

. Then

\sin(2A) = \sin A \cos A + \cos A \sin A = 2\sin A \cos A

and

\cos(2A) = \cos^2 A - \sin^2 A

. The tangent formula gives

\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}

Which pair of special angles should I use?

Any pair works if it gives the angle:

75^\circ = 45^\circ + 30^\circ

15^\circ = 45^\circ - 30^\circ

60^\circ - 45^\circ

105^\circ = 60^\circ + 45^\circ

165^\circ = 120^\circ + 45^\circ

135^\circ + 30^\circ

. Different pairs lead to the same final value.

When is $\tan(A + B)$ undefined?

When

1 - \tan A \tan B = 0

, that is when

\tan A \tan B = 1

. Then

A + B

90^\circ

(or

270^\circ

), where tangent does not exist. Example:

\tan 30^\circ \tan 60^\circ = 1

and

30^\circ + 60^\circ = 90^\circ

Glossary

Sum identity: A formula for a trigonometric function of $A + B$ , e.g. $\sin(A + B) = \sin A \cos B + \cos A \sin B$
Difference identity: A formula for a trigonometric function of $A - B$ , e.g. $\cos(A - B) = \cos A \cos B + \sin A \sin B$
Exact value: A value written with whole numbers, fractions and roots, like $\frac{\sqrt{6} + \sqrt{2}}{4}$ , instead of a rounded decimal
Special angles: $30^\circ$ , $45^\circ$ and $60^\circ$ (and their relatives on the unit circle), whose sine, cosine and tangent are known exactly
Quadrant sign: The sign a trigonometric value has in each quadrant: I all positive, II sine, III tangent, IV cosine

Formula Card

Sine of a sum

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

Keeps the sign, mixes sine and cosine

Sine of a difference

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

Minus between the two products

Cosine of a sum

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

The sign flips to minus

Cosine of a difference

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

The sign flips to plus

Tangent of a sum

\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}

Numerator keeps, denominator flips the sign

Tangent of a difference

\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}

Numerator minus, denominator plus

Teacher Guide: Sum and Difference Identities

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Sine

Cosine

Tangent

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Combining Two Waves

Rotating Points in Games and Robotics

The Angle Between Two Roads

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Sine of a sum

Sine of a difference

Cosine of a sum

Cosine of a difference

Tangent of a sum

Tangent of a difference

More in This Topic

Reciprocal Identities

Pythagorean Identities

Even and Odd Trigonometric Identities

Cofunction Identities