CMClearMathAcademy

Evaluating Trig Values in All Quadrants

A free Trigonometry lesson from the “The Unit Circle” unit, with a worked example and practice problems including step-by-step solutions.

The unit circle records each angle as a coordinate pair (cos theta, sin theta). Exact values, the six trig functions, reciprocal relationships, and quadrant signs all come from that coordinate model.

What you'll learn

Why it matters: Circular motion, waves, direction, and rotation all use the unit circle to convert angles into coordinates.

Worked example

Problem. Evaluating Trig Values in All Quadrants: Evaluate tan(4pi/3).

  1. Tangent is sine divided by cosine.
  2. Use the values at 4pi/3.
  3. tan(4pi/3) = sqrt(3).

Answer: sqrt(3)

Practice problems

1. Evaluating Trig Values in All Quadrants: Give the unit-circle coordinates for 5pi/6.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluating Trig Values in All Quadrants: Give the unit-circle coordinates for 5pi/6.
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. Coordinates are (cos(theta), sin(theta)).
  4. At 5pi/6, the point is (-sqrt(3)/2, 1/2).
  5. Use exact values.
  6. Check the result by substituting or estimating: the response should match (-sqrt(3)/2, 1/2) and make sense in the original problem.

Answer: (-sqrt(3)/2, 1/2)

2. Evaluating Trig Values in All Quadrants: Evaluate sin(7pi/6).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluating Trig Values in All Quadrants: Evaluate sin(7pi/6).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Sine is the y-coordinate.
  4. Use the point (-sqrt(3)/2, -1/2).
  5. sin(7pi/6) = -1/2.
  6. Check the result by substituting or estimating: the response should match -1/2 and make sense in the original problem.

Answer: -1/2

3. Evaluating Trig Values in All Quadrants: Evaluate cos(5pi/4).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Evaluating Trig Values in All Quadrants: Evaluate cos(5pi/4).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Cosine is the x-coordinate.
  4. Use the point (-sqrt(2)/2, -sqrt(2)/2).
  5. cos(5pi/4) = -sqrt(2)/2.
  6. Check the result by substituting or estimating: the response should match -sqrt(2)/2 and make sense in the original problem.

Answer: -sqrt(2)/2

4. Evaluating Trig Values in All Quadrants: Evaluate tan(4pi/3).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluating Trig Values in All Quadrants: Evaluate tan(4pi/3).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Tangent is sine divided by cosine.
  4. Use the values at 4pi/3.
  5. tan(4pi/3) = sqrt(3).
  6. Check the result by substituting or estimating: the response should match sqrt(3) and make sense in the original problem.

Answer: sqrt(3)

5. Evaluating Trig Values in All Quadrants: If sin(theta) = -1/2, find csc(theta).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Evaluating Trig Values in All Quadrants: If sin(theta) = -1/2, find csc(theta).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Cosecant is reciprocal sine.
  4. Take the reciprocal of sine.
  5. csc(theta) = -2.
  6. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

Practice this interactively with instant feedback and an AI tutor.

Practice Evaluating Trig Values in All Quadrants Take the free placement check

More Trigonometry lessons