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AP Precalculus Unit 3B Review – Trigonometric Equations, Identities, and Inverse Functions

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Q1. Solve for if .

Background

Topic: Solving Trigonometric Equations

This question tests your ability to solve trigonometric equations involving double angles and to find all solutions within a specified interval.

Key Terms and Formulas

  • Double Angle Identity:

  • Standard Angles: Solutions should be in terms of .

Step-by-Step Guidance

  1. Rewrite using the double angle identity: .

  2. Set up the equation: .

  3. Bring all terms to one side: .

  4. Factor out : .

Try solving on your own before revealing the answer!

Q2. Solve for

Background

Topic: Solving Quadratic Trigonometric Equations

This question asks you to solve a quadratic equation in terms of and find all solutions in the given interval.

Key Terms and Formulas

  • Quadratic Equation:

  • Trigonometric Values for Standard Angles

Step-by-Step Guidance

  1. Let to rewrite the equation as .

  2. Factor or use the quadratic formula to solve for .

  3. Once you have the values for , set up equations (each solution).

  4. Find all in that satisfy each equation.

Try solving on your own before revealing the answer!

Q3. Solve for

Background

Topic: Solving Trigonometric Equations with Double Angles

This problem involves expressing in terms of and , then solving for .

Key Terms and Formulas

  • Double Angle Identity:

Step-by-Step Guidance

  1. Rewrite as .

  2. Set up the equation: .

  3. Bring all terms to one side: .

  4. Factor where possible or rearrange to isolate or .

Try solving on your own before revealing the answer!

Q4. Solve for

Background

Topic: Solving Trigonometric Equations Involving Secant

This question tests your ability to manipulate equations involving and convert to .

Key Terms and Formulas

Step-by-Step Guidance

  1. Add $5.

  2. Divide both sides by $2\sec x = 2$.

  3. Recall , so .

  4. Solve for and find all in that satisfy this value.

Try solving on your own before revealing the answer!

Q5. Solve for

Background

Topic: Solving Trigonometric Equations

This equation can be solved by factoring and considering when or .

Key Terms and Formulas

  • Zero Product Property

Step-by-Step Guidance

  1. Bring all terms to one side: .

  2. Factor out : .

  3. Set each factor equal to zero: and .

  4. Solve each equation for in .

Try solving on your own before revealing the answer!

Q6. Solve for

Background

Topic: Solving Trigonometric Equations Involving Cosecant

This question involves isolating and converting to .

Key Terms and Formulas

Step-by-Step Guidance

  1. Add $2.

  2. Divide both sides by $3\csc x = -2$.

  3. Recall , so .

  4. Solve for and find all in that satisfy this value.

Try solving on your own before revealing the answer!

Q7. Solve for

Background

Topic: Solving Trigonometric Equations with Multiple Angles

This question tests your understanding of the tangent function and its periodicity.

Key Terms and Formulas

  • General Solution for :

  • Period of Tangent:

Step-by-Step Guidance

  1. Let , so .

  2. Find all such that in (since is in ).

  3. Set for integer .

  4. Solve for by dividing each solution for by $3$.

Try solving on your own before revealing the answer!

Q8. Let and . For which in is ?

Background

Topic: Comparing Trigonometric Functions

This question asks you to solve an inequality involving a trigonometric function.

Key Terms and Formulas

  • Solving inequalities with trigonometric functions

Step-by-Step Guidance

  1. Set up the inequality: .

  2. Add $3.

  3. Divide both sides by $4\cos x < 1$.

  4. Interpret the solution in the context of the interval .

Try solving on your own before revealing the answer!

Q9. Let and . For which in is ?

Background

Topic: Comparing Trigonometric Functions

This question involves solving an inequality between two sine functions.

Key Terms and Formulas

  • Solving inequalities with trigonometric functions

Step-by-Step Guidance

  1. Set up the inequality: .

  2. Subtract from both sides: .

  3. Subtract $1-1 \leq 2 \sin x$.

  4. Divide both sides by $2-\frac{1}{2} \leq \sin x$.

Try solving on your own before revealing the answer!

Q10. For which in is ?

Background

Topic: Solving Trigonometric Inequalities

This question involves solving a quadratic inequality in terms of .

Key Terms and Formulas

  • Quadratic Inequality

Step-by-Step Guidance

  1. Rewrite the inequality: .

  2. Bring all terms to one side: .

  3. Multiply both sides by (reverse the inequality): .

  4. Factor: .

Try solving on your own before revealing the answer!

Q11. Evaluate using the sum/difference formulas.

Background

Topic: Sum and Difference Formulas for Sine

This question tests your ability to use the sine addition formula to evaluate non-standard angles.

Key Terms and Formulas

  • Sum Formula:

Step-by-Step Guidance

  1. Express as a sum of two standard angles (e.g., ).

  2. Apply the sum formula: .

  3. Recall the exact values for , , , and .

  4. Substitute these values into the formula and simplify.

Try solving on your own before revealing the answer!

Q12. Evaluate using the sum/difference formulas.

Background

Topic: Sum and Difference Formulas for Cosine

This question tests your ability to use the cosine addition or subtraction formula for non-standard angles.

Key Terms and Formulas

  • Cosine Sum Formula:

  • Cosine Difference Formula:

Step-by-Step Guidance

  1. Express as a sum or difference of standard angles (e.g., ).

  2. Apply the appropriate formula: .

  3. Recall the exact values for , , , and .

  4. Substitute these values into the formula and simplify.

Try solving on your own before revealing the answer!

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