Guided Practice
Apply your knowledge of the unit circle and trigonometric identities through structured practice problems with step-by-step guidance.
Learn
This lesson provides guided practice to reinforce your understanding of unit circle values and trigonometric identities. Each problem includes hints and solution strategies to help you build confidence.
Key Concepts to Apply
- Unit Circle Coordinates: Remember that for any angle theta, the point on the unit circle is (cos theta, sin theta)
- Reference Angles: Use reference angles to find trig values in all four quadrants
- Pythagorean Identities: sin^2(theta) + cos^2(theta) = 1 and its variations
- Quotient Identities: tan(theta) = sin(theta)/cos(theta)
- Reciprocal Identities: csc, sec, and cot as reciprocals of sin, cos, and tan
Problem-Solving Strategy
- Identify what is given and what you need to find
- Determine which identity or unit circle value applies
- Consider the quadrant to determine the sign (+/-)
- Substitute and simplify
- Verify your answer makes sense
Examples
Work through these guided examples before attempting the practice problems.
Example 1: Finding Exact Values
Problem: Find the exact value of sin(5pi/6).
Step 1: Identify the reference angle. 5pi/6 is in Quadrant II, and its reference angle is pi - 5pi/6 = pi/6.
Step 2: Recall that sin(pi/6) = 1/2.
Step 3: In Quadrant II, sine is positive.
Solution: sin(5pi/6) = 1/2
Example 2: Using Pythagorean Identity
Problem: If cos(theta) = 3/5 and theta is in Quadrant IV, find sin(theta).
Step 1: Use sin^2(theta) + cos^2(theta) = 1.
Step 2: sin^2(theta) + (3/5)^2 = 1, so sin^2(theta) = 1 - 9/25 = 16/25.
Step 3: sin(theta) = +/- 4/5. In Quadrant IV, sine is negative.
Solution: sin(theta) = -4/5
Example 3: Simplifying with Identities
Problem: Simplify: (1 - cos^2(x)) / sin(x)
Step 1: Recognize that 1 - cos^2(x) = sin^2(x) (Pythagorean identity).
Step 2: Substitute: sin^2(x) / sin(x).
Step 3: Simplify by canceling: sin(x).
Solution: sin(x)
Practice
Try these problems on your own. Use the strategies from the examples above.
Problem 1: Find the exact value of cos(7pi/6).
Show Hint
7pi/6 is in Quadrant III. Find the reference angle first.
Problem 2: Find the exact value of tan(3pi/4).
Show Hint
Use tan = sin/cos with the unit circle values for 3pi/4.
Problem 3: If sin(theta) = -5/13 and theta is in Quadrant III, find cos(theta).
Show Hint
Use the Pythagorean identity and remember the sign in Quadrant III.
Problem 4: Simplify: tan(x) * cos(x)
Show Hint
Replace tan(x) with sin(x)/cos(x) and simplify.
Problem 5: Find sec(theta) if cos(theta) = -2/3.
Show Hint
Secant is the reciprocal of cosine.
Problem 6: Simplify: cot(x) * sin(x)
Show Hint
Write cot(x) in terms of sine and cosine.
Problem 7: Find the exact value of sin(11pi/6).
Show Hint
11pi/6 is in Quadrant IV. What is the reference angle?
Problem 8: If tan(theta) = 4/3 and theta is in Quadrant I, find sin(theta) and cos(theta).
Show Hint
Use tan = opposite/adjacent to set up a right triangle, then find the hypotenuse.
Problem 9: Simplify: (sin^2(x) + cos^2(x)) / sec(x)
Show Hint
The numerator equals 1. What is 1/sec(x)?
Problem 10: Find cos(theta) if sec(theta) = -5/4 and theta is in Quadrant II.
Show Hint
Use the reciprocal relationship and verify the sign matches the quadrant.
Check Your Understanding
Test yourself with these review questions to ensure you've mastered the concepts.
Question 1: What is the exact value of sin(2pi/3)?
Question 2: If cos(theta) = 5/13 and theta is in Quadrant I, what is tan(theta)?
Question 3: True or False: sec^2(x) - 1 = tan^2(x)
Question 4: What quadrant has positive sine but negative cosine?
Show Answers
- sqrt(3)/2
- 12/5 (using sin = 12/13 from Pythagorean identity)
- True (this is a Pythagorean identity)
- Quadrant II
Next Steps
- Review any problems that felt challenging
- Create flashcards for unit circle values you haven't memorized
- Move on to Word Problems to apply these skills in context
- Return to practice problems periodically for review