Tangent and the reciprocal trigonometric functions

About this lesson

Precalculus › Trigonometric functions

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Tangent and the reciprocal trigonometric functions

Warm-up

Three quick questions from earlier lessons.

Discover

On the unit circle, the point at angle θ\theta is (cos⁡θ,sin⁡θ)(\cos\theta, \sin\theta). The line from the origin to it has a slope. What is that slope?

  1. sin⁡θcos⁡θ\dfrac{\sin\theta}{\cos\theta}
  2. cos⁡θsin⁡θ\dfrac{\cos\theta}{\sin\theta}

Key idea

Four more functions come from sine and cosine.

tan⁡θ=sin⁡θcos⁡θ\tan\theta = \dfrac{\sin\theta}{\cos\theta}, sec⁡θ=1cos⁡θ\sec\theta = \dfrac{1}{\cos\theta}, csc⁡θ=1sin⁡θ\csc\theta = \dfrac{1}{\sin\theta}, cot⁡θ=cos⁡θsin⁡θ\cot\theta = \dfrac{\cos\theta}{\sin\theta}. Each is undefined where its denominator is 0.

Worked example

Find tan⁡π3\tan\dfrac{\pi}{3} and sec⁡π3\sec\dfrac{\pi}{3}.

tan⁡π3=3/21/2=3\tan\dfrac{\pi}{3} = \dfrac{\sqrt{3}/2}{1/2} = \sqrt{3}
sin(π/3) = √3/2, cos(π/3) = 1/2.
sec⁡π3=11/2=2\sec\dfrac{\pi}{3} = \dfrac{1}{1/2} = 2
sec is 1 over cos.

Your turn, guided

Your turn: θ=π4\theta = \dfrac{\pi}{4}.

tan⁡π4=1\tan\dfrac{\pi}{4} = 1
sin and cos are both √2/2.
csc⁡π4=2\csc\dfrac{\pi}{4} = \sqrt{2}
1 ÷ (√2/2) = 2/√2.

Practice ★

  1. Find the exact value of tan⁡(2π3)\tan\left(\frac{2 \pi}{3}\right).
  2. Find the exact value of tan⁡(π6)\tan\left(\frac{\pi}{6}\right).
  3. Find the exact value of tan⁡(π3)\tan\left(\frac{\pi}{3}\right).
  4. Find the exact value of tan⁡(5π6)\tan\left(\frac{5 \pi}{6}\right).

Watch out!

Practice ★★

  1. Find the exact value of sec⁡(π6)\sec\left(\frac{\pi}{6}\right).
  2. Find the exact value of sec⁡(π3)\sec\left(\frac{\pi}{3}\right).
  1. θ\theta is in the first quadrant and sin⁡θ=35\sin\theta = \frac{3}{5}. Find tan⁡θ\tan\theta.
  2. θ\theta is in the first quadrant and sin⁡θ=725\sin\theta = \frac{7}{25}. Find tan⁡θ\tan\theta.

Exit ticket

Three questions to check you have got it.

  1. Find the exact value of tan⁡(2π3)\tan\left(\frac{2 \pi}{3}\right).
  2. Find the exact value of tan⁡(π6)\tan\left(\frac{\pi}{6}\right).
  3. Find the exact value of tan⁡(π3)\tan\left(\frac{\pi}{3}\right).

Summary card

Tangent and reciprocals

  • tan⁡=sin⁡cos⁡\tan = \dfrac{\sin}{\cos}, cot⁡=cos⁡sin⁡\cot = \dfrac{\cos}{\sin}.
  • sec⁡=1cos⁡\sec = \dfrac{1}{\cos}, csc⁡=1sin⁡\csc = \dfrac{1}{\sin}.

Watch out: sec goes with cos, csc with sin.

End of the lesson

Next lesson: Right-triangle trigonometry Back to the chapter

Answers: Tangent and the reciprocal trigonometric functions

A

  1. −3- \sqrt{3}
  2. 33\frac{\sqrt{3}}{3}
  3. 3\sqrt{3}
  4. −33- \frac{\sqrt{3}}{3}

B

  1. 233\frac{2 \sqrt{3}}{3}
  2. 22

C

  1. 34\frac{3}{4}
  2. 724\frac{7}{24}

D

  1. −3- \sqrt{3}
  2. 33\frac{\sqrt{3}}{3}
  3. 3\sqrt{3}

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