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Reciprocal and Quotient Identities

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

Every reciprocal trig function is just one over a basic one: cosecant is 1/sin, secant is 1/cos, and cotangent is 1/tan. The quotient identities express tangent and cotangent directly from sine and cosine: tan(theta) = sin(theta)/cos(theta) and cot(theta) = cos(theta)/sin(theta). Knowing only sin and cos for an angle lets you build all six functions and rewrite expressions in simpler terms.

What you'll learn

Why it matters: Engineers and physicists often record only sine and cosine from a measurement, then reconstruct secant, cosecant, and cotangent using these identities, for example when computing impedance phase angles or slopes in surveying.

Worked example

Problem. Given sin(theta) = 3/5 and cos(theta) = 4/5, find tan(theta).

  1. Use the quotient identity tan(theta) = sin(theta)/cos(theta).
  2. Substitute: tan(theta) = (3/5)/(4/5).
  3. Dividing, the fifths cancel: tan(theta) = 3/4.

Answer: 3/4

Practice problems

1. If sin(theta) = 1/4, find csc(theta).

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

Answer: 4

2. If cos(theta) = 1/3, find sec(theta).

Show solution
  1. Warm-up: First identify exactly what the question is asking: If cos(theta) = 1/3, find sec(theta).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Secant is the reciprocal of cosine: sec(theta) = 1/cos(theta).
  4. Take the reciprocal of 1/3.
  5. sec(theta) = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Which identity correctly defines cotangent in terms of sine and cosine?

Choices: cot(theta) = sin(theta)/cos(theta) · cot(theta) = cos(theta)/sin(theta) · cot(theta) = 1/sin(theta) · cot(theta) = cos(theta) * sin(theta)

Show solution
  1. Cotangent is the reciprocal of tangent, and tan = sin/cos.
  2. Flipping sin/cos gives cos/sin.
  3. So cot(theta) = cos(theta)/sin(theta).

Answer: cot(theta) = cos(theta)/sin(theta)

4. Given sin(theta) = 3/5 and cos(theta) = 4/5, find cot(theta).

Show solution
  1. Use the quotient identity cot(theta) = cos(theta)/sin(theta).
  2. Substitute: cot(theta) = (4/5)/(3/5).
  3. The fifths cancel: cot(theta) = 4/3.

Answer: 4/3

5. If tan(theta) = 5/2, find cot(theta).

Show solution
  1. Core Practice: First identify exactly what the question is asking: If tan(theta) = 5/2, find cot(theta).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Cotangent is the reciprocal of tangent: cot(theta) = 1/tan(theta).
  4. Flip 5/2.
  5. cot(theta) = 2/5.
  6. Check the result by substituting or estimating: the response should match 2/5 and make sense in the original problem.

Answer: 2/5

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