Precalculus Trigonometric Values, Identities, and Equations Study Guide

Q1. Find the exact value of cos(7π/12)

Background

Topic: Trigonometric Values of Nonstandard Angles

This question tests your ability to use sum and difference identities to find the exact value of trigonometric functions for angles that are not standard (like 30°, 45°, 60°, etc.).

Key Terms and Formulas

• Sum and Difference Formulas for Cosine:

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

Step-by-Step Guidance

1. Express $7\pi/12$ as a sum or difference of angles with known values (for example, $\pi/3$ and $\pi/4$).

2. Write $\cos(7\pi/12)$ using the sum or difference formula for cosine.

3. Substitute the known values for $\cos$ and $\sin$ of the chosen angles.

4. Simplify the expression as much as possible, but stop before the final calculation.

Try solving on your own before revealing the answer!

Q2. Find the exact value of sin(-15°)

Background

Topic: Trigonometric Values of Negative Angles

This question tests your understanding of the sine function for negative angles and how to use sum/difference identities or symmetry properties.

Key Terms and Formulas

• Odd-Even Properties: $\sin(-x) = -\sin(x)$

• Sum and Difference Formula for Sine:

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

Step-by-Step Guidance

1. Use the odd property of sine to rewrite $\sin(-15^\circ)$ as $-\sin(15^\circ)$.

2. Express $15^\circ$ as $45^\circ - 30^\circ$ to use the difference formula.

3. Apply the sine difference formula to $\sin(45^\circ - 30^\circ)$.

4. Substitute the known values for $\sin$ and $\cos$ of $45^\circ$ and $30^\circ$.

Try solving on your own before revealing the answer!

Q3. Find the exact value of tan(105°)

Background

Topic: Tangent of Nonstandard Angles

This question tests your ability to use sum and difference identities for tangent to find the exact value for angles like $105^\circ$.

Key Terms and Formulas

• Sum Formula for Tangent:

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

Step-by-Step Guidance

1. Express $105^\circ$ as a sum of two standard angles (for example, $60^\circ + 45^\circ$).

2. Apply the tangent sum formula to $\tan(60^\circ + 45^\circ)$.

3. Substitute the known values for $\tan 60^\circ$ and $\tan 45^\circ$.

4. Simplify the expression, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q4. Given $\cos \alpha = \frac{3}{5}$, $0 < \alpha < \frac{\pi}{2}$ and $\sin \beta = -\frac{4}{5}$, $-\frac{\pi}{2} < \beta < 0$, find:

Background

Topic: Trigonometric Values from Given Information

This question tests your ability to use given trigonometric values and quadrant information to find other trigonometric values and to use sum and difference identities.

Key Terms and Formulas

• Pythagorean Identity: $\sin^2 x + \cos^2 x = 1$

• Sum and Difference Formulas:

$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$

$\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

Step-by-Step Guidance

1. Use the Pythagorean identity to find $\sin \alpha$ and $\cos \beta$ (pay attention to the signs based on the quadrants).

2. For part (a), use the cosine sum formula to set up $\cos(\alpha + \beta)$.

3. For part (b), find $\tan \alpha$ and $\tan \beta$ using $\sin$ and $\cos$ values, then set up the tangent difference formula.

4. Simplify the expressions as much as possible, but stop before the final calculation.

Try solving on your own before revealing the answer!

Q5. Find the exact value of $\tan\left(\frac{5\pi}{8}\right)$

Background

Topic: Tangent of Nonstandard Angles (Radians)

This question tests your ability to use sum or difference identities for tangent with radian measures.

Key Terms and Formulas

• Sum or Difference Formula for Tangent:

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Step-by-Step Guidance

1. Express $\frac{5\pi}{8}$ as a sum or difference of angles with known tangent values (for example, $\frac{\pi}{4}$ and $\frac{\pi}{8}$).

2. Apply the tangent sum or difference formula.

3. Substitute the known values for the tangents of the chosen angles.

4. Simplify the expression, but do not compute the final value yet.

Try solving on your own before revealing the answer!

Q6. Given $\cos \theta = -\frac{3}{5}$, $\frac{\pi}{2} < \theta < \pi$, find:

Background

Topic: Half-Angle and Double-Angle Formulas

This question tests your ability to use half-angle and double-angle identities, as well as quadrant information, to find exact trigonometric values.

Key Terms and Formulas

• Half-Angle Formulas:

$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$

$\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$

$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta}$

Step-by-Step Guidance

1. Use the Pythagorean identity to find $\sin \theta$ (pay attention to the sign based on the quadrant).

2. For part (a), use the half-angle formula for sine, and determine the correct sign based on the interval for $\frac{\theta}{2}$.

3. For part (b), use the double-angle formula for cosine, substituting the known value for $\cos \theta$.

4. For part (c), use the half-angle formula for tangent, substituting the values you have found.

Try solving on your own before revealing the answer!

Q7. Establish the identity: $3\sin^2\theta + 4\cos^2\theta = 3 + \cos^2\theta$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to manipulate and verify trigonometric identities using algebraic and Pythagorean identities.

Key Terms and Formulas

• Pythagorean Identity: $\sin^2\theta + \cos^2\theta = 1$

Step-by-Step Guidance

1. Start with the left side: $3\sin^2\theta + 4\cos^2\theta$.

2. Use the Pythagorean identity to rewrite $\sin^2\theta$ in terms of $\cos^2\theta$.

3. Simplify the expression and compare it to the right side, but stop before the final verification.

Try solving on your own before revealing the answer!

Q8. Establish the identity: $\frac{1 + \sin \beta}{1 - \sin \beta} = \frac{\csc \beta + 1}{\csc \beta - 1}$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to manipulate trigonometric expressions and use reciprocal identities.

Key Terms and Formulas

• Reciprocal Identity: $\csc \beta = \frac{1}{\sin \beta}$

Step-by-Step Guidance

1. Rewrite $\csc \beta$ in terms of $\sin \beta$ in the right-hand side expression.

2. Simplify the numerator and denominator to get a single fraction.

3. Compare the simplified right side to the left side, but stop before the final verification.

Try solving on your own before revealing the answer!

Q9. Establish the identity: $\frac{\cos \theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{\cos \theta} = 2\sec \theta$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to combine fractions and use reciprocal identities.

Key Terms and Formulas

• Reciprocal Identity: $\sec \theta = \frac{1}{\cos \theta}$

Step-by-Step Guidance

1. Find a common denominator for the two fractions on the left side.

2. Combine the fractions into a single expression.

3. Simplify the numerator and denominator, and relate the result to $2\sec \theta$, but stop before the final verification.

Try solving on your own before revealing the answer!

Q10. Establish the identity: $\frac{1 + \tan \theta}{1 + \cot \theta} = \tan \theta$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to rewrite tangent and cotangent in terms of sine and cosine and simplify.

Key Terms and Formulas

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

• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Step-by-Step Guidance

1. Rewrite $\tan \theta$ and $\cot \theta$ in terms of sine and cosine.

2. Combine the numerator and denominator into single fractions.

3. Simplify the complex fraction, but stop before the final verification.

Try solving on your own before revealing the answer!

Q11. Establish the identity: $\sec \mu - \tan \mu = \frac{\cos \mu}{1 + \sin \mu}$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to manipulate trigonometric expressions using reciprocal and quotient identities.

Key Terms and Formulas

• $\sec \mu = \frac{1}{\cos \mu}$

• $\tan \mu = \frac{\sin \mu}{\cos \mu}$

Step-by-Step Guidance

1. Rewrite $\sec \mu$ and $\tan \mu$ in terms of sine and cosine.

2. Combine the terms over a common denominator.

3. Simplify the numerator and denominator, but stop before the final verification.

Try solving on your own before revealing the answer!

Q12. Establish the identity: $\frac{\tan A + \tan B}{\cot A + \cot B} = \tan A \tan B$

Background

Topic: Verifying Trigonometric Identities

This question tests your ability to manipulate and combine tangent and cotangent expressions.

Key Terms and Formulas

• $\tan x = \frac{\sin x}{\cos x}$

• $\cot x = \frac{\cos x}{\sin x}$

Step-by-Step Guidance

1. Rewrite all tangents and cotangents in terms of sine and cosine.

2. Combine the numerator and denominator into single fractions.

3. Simplify the complex fraction, but stop before the final verification.

Try solving on your own before revealing the answer!

Q13. Solve $2\sin \theta + 3 = 2$ for $0 \leq \theta < 2\pi$

Background

Topic: Solving Basic Trigonometric Equations

This question tests your ability to solve for $\theta$ in a basic sine equation within a specified interval.

Key Terms and Formulas

• Inverse Sine Function: $\sin^{-1}(x)$

Step-by-Step Guidance

1. Isolate $\sin \theta$ on one side of the equation.

2. Set up the equation $\sin \theta =$ (some value).

3. Determine all solutions for $\theta$ in the interval $[0, 2\pi)$, but stop before listing the final values.

Try solving on your own before revealing the answer!

Q14. Solve $4\cos^2 \theta - 3 = 0$ for $0 \leq \theta < 2\pi$

Background

Topic: Solving Quadratic Trigonometric Equations

This question tests your ability to solve quadratic equations in terms of cosine and find all solutions in a given interval.

Key Terms and Formulas

• Quadratic Equation

• Inverse Cosine Function: $\cos^{-1}(x)$

Step-by-Step Guidance

1. Isolate $\cos^2 \theta$ and solve for $\cos \theta$.

2. Set up the equation $\cos \theta =$ (some value).

3. Determine all solutions for $\theta$ in the interval $[0, 2\pi)$, but stop before listing the final values.

Try solving on your own before revealing the answer!

Q15. Solve $\cos 2\theta - \sin^2 \theta + \sin \theta = 0$ for $0 \leq \theta < 2\pi$

Background

Topic: Solving Trigonometric Equations with Multiple Angles

This question tests your ability to use double-angle identities and solve trigonometric equations involving multiple angles.

Key Terms and Formulas

• Double-Angle Identity: $\cos 2\theta = 1 - 2\sin^2 \theta$

Step-by-Step Guidance

1. Rewrite $\cos 2\theta$ using the double-angle identity in terms of $\sin \theta$.

2. Combine like terms to form a quadratic equation in $\sin \theta$.

3. Solve for $\sin \theta$, but stop before finding the final values for $\theta$.

Try solving on your own before revealing the answer!

Q16. Solve $\tan\left(\frac{\theta}{2} + \frac{\pi}{3}\right) = 1$ for $0 \leq \theta < 2\pi$

Background

Topic: Solving Tangent Equations with Multiple Angles

This question tests your ability to solve equations involving tangent and angle addition.

Key Terms and Formulas

• General Solution for $\tan x = 1$: $x = \frac{\pi}{4} + n\pi$

Step-by-Step Guidance

1. Set $\frac{\theta}{2} + \frac{\pi}{3} = \frac{\pi}{4} + n\pi$ for integer $n$.

2. Solve for $\theta$ in terms of $n$.

3. Find all solutions for $\theta$ in $[0, 2\pi)$, but stop before listing the final values.

Try solving on your own before revealing the answer!

Q17. Give a general formula for all solutions to $\sin(2\theta) = -1$

Background

Topic: General Solutions to Trigonometric Equations

This question tests your ability to find all solutions to a sine equation using the periodicity of the sine function.

Key Terms and Formulas

• General Solution for $\sin x = -1$: $x = \frac{3\pi}{2} + 2\pi n$

Step-by-Step Guidance

1. Set $2\theta = \frac{3\pi}{2} + 2\pi n$ for integer $n$.

2. Solve for $\theta$ in terms of $n$.

Try solving on your own before revealing the answer!

Q18. Give a general formula for all solutions to $\tan\left(\frac{\theta}{2}\right) = -1$

Background

Topic: General Solutions to Tangent Equations

This question tests your ability to find all solutions to a tangent equation using the periodicity of the tangent function.

Key Terms and Formulas

• General Solution for $\tan x = -1$: $x = -\frac{\pi}{4} + n\pi$

Step-by-Step Guidance

1. Set $\frac{\theta}{2} = -\frac{\pi}{4} + n\pi$ for integer $n$.

2. Solve for $\theta$ in terms of $n$.

Try solving on your own before revealing the answer!

Q19. Solve $\cos \theta + \sin \theta = 0$ for $0 \leq \theta < 2\pi$

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve equations involving both sine and cosine.

Key Terms and Formulas

• Algebraic Manipulation

• Reference Angles

Step-by-Step Guidance

1. Isolate one trigonometric function (e.g., $\cos \theta = -\sin \theta$).

2. Divide both sides by $\cos \theta$ (if appropriate) to get $1 = -\tan \theta$.

3. Solve for $\theta$ in $[0, 2\pi)$, but stop before listing the final values.

Try solving on your own before revealing the answer!

Q20. Solve $\sin(2\theta)\sin \theta = \cos \theta$ for $0 \leq \theta < 2\pi$

Background

Topic: Solving Trigonometric Equations with Products

This question tests your ability to use product-to-sum identities and solve trigonometric equations.

Key Terms and Formulas

• Double-Angle Identity: $\sin(2\theta) = 2\sin \theta \cos \theta$

Step-by-Step Guidance

1. Rewrite $\sin(2\theta)$ using the double-angle identity.

2. Expand and rearrange the equation to collect like terms.

3. Factor the equation and solve for $\theta$ in $[0, 2\pi)$, but stop before listing the final values.

Try solving on your own before revealing the answer!