Apply fundamental and double-angle trigonometric identities

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Geometry and Trigonometry

Using familiar trigonometric relationships to rewrite and evaluate expressions

An identity is an equality that is true for every value in its stated domain. Trigonometric identities let you rewrite an expression without changing its value. This lesson focuses on the fundamental relationships between sine, cosine, and tangent, and on formulas for the sine, cosine, and tangent of twice an angle. The aim is not simply to remember formulas: it is to recognize which form makes a calculation or simplification easier. Angles may be measured in degrees or radians; keep the chosen unit consistent, particularly when using a calculator.

What you will learn

Use the fundamental identity relating sine and cosine, and the definition of tangent.
Apply the sine, cosine, and tangent double-angle identities.
Choose an equivalent identity that fits the information given or the expression to be simplified.
Check identity-based results using domains, exact values, and graphing technology.

1. Prior knowledge: sine, cosine, and tangent

For an angle

\theta

, sine, cosine, and tangent are written

\sin\theta

,

\cos\theta

, and

\tan\theta

. On a right-angled triangle, sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. The quotient relationship between tangent and the other two functions is useful even when working with angles beyond a right triangle.

A fundamental identity is

\sin^2\theta+\cos^2\theta=1

. The notation

\sin^2\theta

means

(\sin\theta)^2

, not

\sin(\theta^2)

. The identity applies for every real angle.

Tangent is defined by

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

wherever

\cos\theta\ne 0

. This restriction matters: tangent is undefined when the denominator is zero. Rearranging the fundamental identity and dividing by

\cos^2\theta

also gives

1+\tan^2\theta=\frac{1}{\cos^2\theta}

where

\cos\theta\ne0

. This form can be useful when an expression contains tangent.

\sin^2\theta+\cos^2\theta=1,\qquad \tan\theta=\frac{\sin\theta}{\cos\theta}

The fundamental sine-cosine identity is valid for all real angles.
The quotient definition of tangent requires a nonzero cosine.
Squaring applies to the function value: $\sin^2\theta=(\sin\theta)^2$ .

2. The double-angle identities

A double angle means twice a given angle, written

2\theta

. The angle-addition relationships for sine and cosine lead to formulas for

\sin(2\theta)

and

\cos(2\theta)

. In these formulas, the same angle

\theta

appears in each factor.

The cosine double-angle identity has three equivalent forms. The first uses both sine and cosine. Replacing

\sin^2\theta

using

\sin^2\theta=1-\cos^2\theta

gives a form involving cosine alone; replacing

\cos^2\theta

gives one involving sine alone. Choose the form that matches the information you have.

For tangent, divide the sine double-angle formula by the cosine double-angle formula and use

\tan\theta=\sin\theta/\cos\theta

. This gives a compact expression, but its denominator must not be zero, and the original tangent must be defined. Check these restrictions when using the formula.

These are algebraic relationships, not estimates. For a specified angle, calculate the original expression and its identity-based form to see that the values agree. On graphs, the curves for the two sides of an identity lie on top of each other wherever both sides are defined.

\sin(2\theta)=2\sin\theta\cos\theta,\quad \cos(2\theta)=\cos^2\theta-\sin^2\theta=1-2\sin^2\theta=2\cos^2\theta-1,\quad \tan(2\theta)=\frac{2\tan\theta}{1-\tan^2\theta}

The sine double-angle formula has one standard form.
The cosine double-angle formula can be written in three equivalent ways.
The tangent double-angle formula is valid only where its terms are defined.

3. Choosing a representation and checking results

Symbolically, use identities to change the form of an expression while preserving its value. Numerically, substitute an angle and evaluate both forms, keeping sufficient calculator precision until the final answer. A small difference in displayed decimal values may result from rounding, not a failure of the identity.

Graphing technology can provide a useful check. Enter the two sides of an identity as separate functions of the same angle, and use the same angle mode and viewing window. If one side is undefined at some angles, a graph may show a gap or fail to draw there; that does not disprove the identity on its valid domain.

In a context, an angle might describe a direction or a repeated rotation. If the original angle is

\theta

, a double-angle expression concerns

2\theta

, in the same unit. Identities help evaluate or rewrite the expression, but they do not change the meaning or units of the angle.

A reliable exam approach is to identify the information given, select a matching identity, substitute carefully, and simplify using ordinary algebra. Finish by checking restrictions and, when appropriate, checking the result numerically. A calculator supports the reasoning; it does not replace showing the identity and algebra used.

Choose a double-angle form based on which trigonometric value is known.
Use graphing technology as a check and account for undefined points.
Keep angle units consistent and preserve exact values when possible.

Worked example

1. Evaluate a double-angle expression exactly

Given

\sin\theta=\frac{3}{5}

and that

\theta

is acute, find

\cos(2\theta)

exactly.

Find the missing trigonometric value
Because $\theta$ is acute, its cosine is positive. Use the fundamental identity to determine cosine from the given sine.
$\cos^2\theta=1-\left(\frac{3}{5}\right)^2=\frac{16}{25},\qquad \cos\theta=\frac{4}{5}$
Select a suitable cosine form
Both sine and cosine are now known, so use the form involving their squares. This avoids first calculating the doubled angle.
$\cos(2\theta)=\cos^2\theta-\sin^2\theta$
Substitute and simplify
Substitute the exact values and subtract the fractions with the same denominator.
$\cos(2\theta)=\left(\frac{4}{5}\right)^2-\left(\frac{3}{5}\right)^2=\frac{7}{25}$

Answer:

\cos(2\theta)=\frac{7}{25}

.

Check: The result is between

-1

and

1

, as a cosine value must be. The acute-angle condition selected the positive value of

\cos\theta

.

Worked example

2. Rewrite an expression using double-angle identities

Rewrite

2\sin x\cos x+\cos^2x-\sin^2x

as a sum of trigonometric functions of the doubled angle.

Recognize the sine double angle
The product term matches the formula for the sine of twice the angle.
$2\sin x\cos x=\sin(2x)$
Recognize the cosine double angle
The difference of the squared cosine and squared sine matches the cosine double-angle formula.
$\cos^2x-\sin^2x=\cos(2x)$
Combine the rewritten terms
Apply both identities to the original expression. The result is a sum of sine and cosine functions of the same doubled angle.
$2\sin x\cos x+\cos^2x-\sin^2x=\sin(2x)+\cos(2x)$

Answer: The expression rewrites as

\sin(2x)+\cos(2x)

.

Check: At

x=0

, the original expression equals

1

and the rewritten expression is

\sin 0+\cos 0=1

.

Worked example

3. Apply the tangent double-angle identity

Given

\tan\theta=\frac{1}{3}

, find

\tan(2\theta)

, assuming the angles are in radians.

Check the formula's denominator
The tangent double-angle formula requires $1-\tan^2\theta\ne0$ . With the given value, the denominator is nonzero.
$1-\left(\frac{1}{3}\right)^2=\frac{8}{9}\ne0$
Substitute into the identity
Use the given tangent in the numerator and denominator of the double-angle formula.
$\tan(2\theta)=\frac{2\left(\frac{1}{3}\right)}{1-\left(\frac{1}{3}\right)^2}$
Simplify the fraction
Divide by multiplying by the reciprocal of the denominator.
$\tan(2\theta)=\frac{2}{3}\cdot\frac{9}{8}=\frac{3}{4}$

Answer:

\tan(2\theta)=\frac{3}{4}

.

Check: As a numerical check, an angle with tangent

1/3

is approximately

0.322

radians. Doubling gives approximately

0.644

radians, whose tangent is approximately

0.750

.

Common mistakes and how to avoid them

Writing

\sin(2\theta)=2\sin\theta

.

Correction: The sine double-angle formula includes both sine and cosine:

\sin(2\theta)=2\sin\theta\cos\theta

.

Treating

\sin^2\theta

as

\sin(\theta^2)

.

Correction: The square is applied after evaluating sine:

\sin^2\theta=(\sin\theta)^2

.

Using

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

without checking whether the denominator is zero.

Correction: Check

1-\tan^2\theta\ne0

, and make sure the tangent values involved are defined.

Assuming a rounded calculator check proves an identity.

Correction: Show the identity and algebraic reasoning. Use numerical or graphical checks only as supporting evidence.

Lesson summary

Use $\sin^2\theta+\cos^2\theta=1$ to relate sine and cosine.
Use $\tan\theta=\sin\theta/\cos\theta$ only where cosine is nonzero.
Apply the sine and cosine double-angle identities, choosing the form that best fits the known information.
Check domain restrictions, keep angle units consistent, and use technology to verify rather than replace reasoning.

Check your understanding

Question 1

If

\sin\theta=\frac{5}{13}

and

\theta

is acute, what is

\cos(2\theta)

?

$\frac{119}{169}$
$\frac{144}{169}$
$-\frac{119}{169}$
$\frac{10}{13}$

Show answer and explanation

\frac{119}{169}

The fundamental identity gives

\cos\theta=12/13

. Therefore

\cos(2\theta)=\cos^2\theta-\sin^2\theta=(144-25)/169=119/169

.

Question 2

Which expression equals

\sin(2x)

?

$2\sin x\cos x$
$\sin^2x+\cos^2x$
$2\sin x$
$\cos^2x-\sin^2x$

Show answer and explanation

2\sin x\cos x

The sine double-angle identity is

\sin(2x)=2\sin x\cos x

. The last option is

\cos(2x)

.

Question 3

Given

\tan\theta=\frac{1}{2}

, what is

\tan(2\theta)

?

$\frac{4}{3}$
$\frac{1}{3}$
$\frac{2}{3}$
$\frac{3}{4}$

Show answer and explanation

\frac{4}{3}

Substitute into the tangent double-angle identity:

\tan(2\theta)=\frac{1}{1-1/4}=4/3

. The denominator is nonzero.

Key terms

Identity: An equality that is true for every value in its stated domain.
Double angle: An angle twice another angle, written $2\theta$ .
Domain restriction: A condition describing values for which an expression is defined, such as requiring a denominator to be nonzero.

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