Introduction

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable. They form a foundational part of precalculus, making them essential for understanding more advanced math concepts. These identities help simplify expressions and solve equations, offering shortcuts that can simplify many trigonometric problems. This article aims to provide a comprehensive trig identities cheat sheet and accompanying practice problems to hone skills in these areas.

Understanding Trigonometric Identities

Trigonometric identities simplify complex expressions and solve trigonometric equations efficiently. Here are three main categories you’ll encounter:

• Pythagorean identities
• Sum and difference identities
• Double angle identities

Each type offers unique tools for tackling a wide range of precalculus problems.

Pythagorean Identities

Pythagorean identities arise from the Pythagorean theorem applied to trigonometry. The main identity is:

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

This fundamental identity can be rearranged to derive other useful forms.

Example:

To derive \tan^2 \theta = \sec^2 \theta - 1, follow these steps:

1. Start from the identity: \sin^2 \theta + \cos^2 \theta = 1.
2. Divide the entire equation by \cos^2 \theta: \frac{\sin^2 \theta}{\cos^2 \theta} + 1 = \frac{1}{\cos^2 \theta}
3. This simplifies to: \tan^2 \theta + 1 = \sec^2 \theta
4. Rearrange it to find: \tan^2 \theta = \sec^2 \theta - 1

Practice Problem:

Prove that \cot^2 \theta + 1 = \csc^2 \theta using \sin^2 \theta + \cos^2 \theta = 1. Hint: Start by dividing the entire equation by \sin^2 \theta

Sum and Difference Identities

Sum and difference identities expand trigonometric expressions, making them essential for solving various problems:

• Sine sum identity: \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta
• Cosine sum identity: \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta

Example:

Expand \sin(30^\circ + 45^\circ) using the sum identity:

1. Identify: \alpha = 30^\circ, \beta = 45^\circ.
2. Apply the sine sum identity: \sin(30^\circ + 45^\circ) = \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ
3. Substitute values: = \frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}
4. Simplify: = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}

Practice Problem:

Use the cosine sum identity to expand \cos(60^\circ + 30^\circ). Solution: \cos(60^\circ)\cos(30^\circ)-\sin(60^\circ)\sin(30^\circ)=0

Double Angle Identities

Double angle identities simplify expressions where angles are doubled:

• Sine double angle: \sin(2\theta) = 2\sin \theta \cos \theta
• Cosine double angle: \cos(2\theta) = \cos^2 \theta - \sin^2 \theta

Example:

Find \sin(2 \cdot 30^\circ) using the double angle identity:

1. Identify: \theta = 30^\circ.
2. Apply the identity: \sin(2 \cdot 30^\circ) = 2 \sin 30^\circ \cos 30^\circ
3. Substitute values: = 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{3}}{2}
4. Simplify: = \frac{\sqrt{3}}{2}

Practice Problem:

Calculate \cos(2 \cdot 45^\circ) using the cosine double angle identity. Solution: \cos^2 (45^\circ) - \sin^2 (45^\circ)=0

Solving Trigonometric Equations

Solving equations with trigonometric identities can simplify and make problems more manageable. The following approach helps:

1. Identify applicable identities to transform the equation.
2. Substitute and simplify using these identities.
3. Solve the simplified equation.

Example:

Solving Trigonometric Equations Using Identities

Solving equations with trigonometric identities can simplify and make problems more manageable. The following approach helps:

• Identify applicable identities to transform the equation.
• Substitute and simplify using these identities.
• Solve the simplified equation.

Example:

Solve for x in the interval 0 \leq x < 2\pi :

\sin^2(x) + \cos(x) = 1

Step 1: Using the Pythagorean identity: \sin^2(x) = 1 - \cos^2(x) , rewrite the equation:

(1 - \cos^2(x)) + \cos(x) = 1

Step 2: Simplify the Equation

1 - \cos^2(x) + \cos(x) = 1

Subtract 1 from both sides:

-\cos^2(x) + \cos(x) = 0

Factor out \cos(x) :

\cos(x) (-\cos(x) + 1) = 0

Step 3: Solve for x

Set each factor equal to zero:

\cos(x) = 0 or -\cos(x) + 1 = 0

For \cos(x) = 0 :

x = \frac{\pi}{2}, \frac{3\pi}{2}

For -\cos(x) + 1 = 0 :

\cos(x) = 1 , so x = 0

Step 4: Express the Final Solution

x = 0, \frac{\pi}{2}, \frac{3\pi}{2}

VII. Quick Reference Chart

Here’s a handy reference chart summarizing key trigonometric identities:

Identity Type	Identity / Formula	Description
Pythagorean Identity	\sin^2 \theta + \cos^2 \theta = 1	Relates sine and cosine.
Sine Sum Identity	\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta	Expands sine of a sum.
Cosine Sum Identity	\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta	Expands cosine of a sum.
Sine Double Angle	\sin(2\theta) = 2\sin \theta \cos \theta	Simplifies sine of double angles.
Cosine Double Angle	\cos(2\theta) = \cos^2 \theta - \sin^2 \theta	Simplifies cosine of double angles.

Conclusion

In conclusion, understanding trigonometric identities is crucial for excelling in precalculus. By consistently reviewing and practicing these concepts, mastery becomes attainable, allowing confidence in tackling any problem assessments may present.

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