Pre-Calculus Trig Worksheet- Practice Problems and Answers

What This Worksheet Actually Covers

Pre-calculus trig worksheets focus on the functions and identities you'll need before hitting calculus. If you're struggling with sine, cosine, and tangent at angles that aren't on the unit circle, this is your starting point.

These problems bridge basic trigonometry and the advanced material coming in calculus. You need to be solid on:

• The six trig functions and their reciprocal relationships
• Radians versus degrees
• Angle addition and subtraction formulas
• Double-angle and half-angle identities
• Solving trigonometric equations
• Graphing trig functions and understanding amplitude, period, and phase shift

The Core Practice Problems

Problem Set 1: Basic Function Evaluation

These test whether you can evaluate trig functions at standard angles. No calculator allowed on these.

Problem 1: Find sin(60°), cos(60°), and tan(60°).

Answer: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3

Problem 2: Evaluate csc(π/3), sec(π/4), and cot(π/6).

Answer: csc(π/3) = 2√3/3, sec(π/4) = √2, cot(π/6) = √3

Problem 3: If cos(θ) = 4/5 and θ is in Quadrant IV, find sin(θ) and tan(θ).

Answer: sin(θ) = -3/5, tan(θ) = -3/4

Problem Set 2: Verifying Trig Identities

These require algebraic manipulation and knowing which identity to apply.

Problem 4: Verify that sin²(x) + cos²(x) = 1.

Answer: This is the Pythagorean identity. It holds for all x by definition of the unit circle.

Problem 5: Simplify (1 - cos²(x)) / sin(x).

Answer: sin²(x)/sin(x) = sin(x)

Problem 6: Verify: (1 + tan²(x)) = sec²(x)

Answer: Using sin²(x) + cos²(x) = 1, divide both sides by cos²(x): tan²(x) + 1 = sec²(x). Verified.

Problem Set 3: Angle Addition and Subtraction

Problem 7: Find cos(75°) using angle addition.

Answer: cos(75°) = cos(45° + 30°) = cos(45°)cos(30°) - sin(45°)sin(30°) = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4

Problem 8: Find sin(15°) using angle subtraction.

Answer: sin(15°) = sin(45° - 30°) = sin(45°)cos(30°) - cos(45°)sin(30°) = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4

Problem Set 4: Double-Angle Problems

Problem 9: If sin(x) = 3/5 and x is in Quadrant II, find sin(2x) and cos(2x).

Answer: cos(x) = -4/5. Then sin(2x) = 2sin(x)cos(x) = 2(3/5)(-4/5) = -24/25. cos(2x) = cos²(x) - sin²(x) = 16/25 - 9/25 = 7/25.

Problem 10: Express cos(4x) in terms of cos(x) only.

Answer: cos(2θ) = 2cos²(θ) - 1, so cos(4x) = 2cos²(2x) - 1 = 2(2cos²(x) - 1)² - 1 = 8cos⁴(x) - 8cos²(x) + 1

Problem Set 5: Solving Trig Equations

Problem 11: Solve 2sin(x) - 1 = 0 for 0 ≤ x < 2π.

Answer: sin(x) = 1/2. Solutions: x = π/6 and x = 5π/6

Problem 12: Solve cos²(x) - 1 = 0 for 0 ≤ x < 2π.

Answer: cos²(x) = 1, so cos(x) = ±1. Solutions: x = 0, π, 2π

Quick Reference: Key Identities

Don't memorize everything. Know these categories:

How to Use These Problems Effectively

Don't just read the answers. That's worthless.

Step 1: Cover the answers. Work each problem cold.

Step 2: If you get stuck, check which identity category applies. Most errors come from trying the wrong identity.

Step 3: Verify your answer by plugging it back in or checking against a known value.

Step 4: If a problem takes more than 5 minutes, check the answer and understand why. Move on. Come back later and try again without looking.

Common Mistakes to Avoid

• Signs: Quadrant matters. Check the sign of your answer before finalizing.
• Radians vs degrees: Pick one system and stick with it. Don't mix them.
• Reciprocal functions: csc, sec, and cot are 1/sin, 1/cos, and 1/tan respectively. Don't confuse them.
• Double-angle formula selection: cos(2x) has three forms. Use the one that matches what you know.

Where to Go From Here

Once you can work through these problems without hesitation, move to:

• Inverse trig functions and their domains
• Law of Sines and Law of Cosines applications
• Parametric equations and polar coordinates
• Limits involving trig functions

These are the actual prerequisites for calculus. The rest is just preparation.