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Graphs of Tangent and Reciprocal Functions

Learning Targets

You should be able to

Concepts / Definitions

y = cscx and sinx y  = secx and cosx y = cotx and tanx


Example 1

Graph $f(\theta) = 4\sec 2\theta +1$


Example 2

Solve $\cot x = -\sqrt 3$ on $(-2\pi, 2\pi)$ (no calculator)



In Class

No Calculator

  1. Convert 300° to radians.
  2. Convert $\frac{4\pi}{3}$ radians to degrees.
  3. What quadrant does 7 radians lie in?
  4. Name a co-terminal angle to $\frac \pi 4$
  5. Evaluate $\sin 1.5\pi$
  6. What is the reference angle for $-\frac{9\pi}{4}$?
  7. If $\sin(-t) = \frac 79$, then $\sin t = ?$
  8. If $\cos\theta = \frac{\sqrt 3}{2}$, then $\cos(\theta-\pi) = ?$
  9. If $\sin\theta = -\frac{\sqrt 3}{2}$, find $\theta$ on $[0,2\pi)$
  10. If $\tan\theta = 0$, then find $\theta$ on $[0,2\pi)$
  11. If $\sec\theta = 2$, then find $\theta$ on $[0,2\pi)$
  12. Find the coordinate on the unit circle that corresponds to an angle of $t = -\frac{5\pi}{6}$
  13. If $\sin = -\frac{\sqrt 2}{2}$ and $\cos x>0$, then $\cos x = ?$
  14. Evaluate $\tan \frac{2\pi}{3}$ and $\sec\frac \pi 3$
  15. Graph $f(x) = 2\sin (2x-\pi)-3$


  1. Convert 1.485 radians to D°M’S” form.
  2. What is the radius of a circle whose arc length is 20 cm and whose angle is 60°?
  3. The radius of a disc is 6 cm. Find the linear speed of a point on the circumference if the disc is rotating at 600 revolutions per min.
  4. A bicycle is moving at a rate of 12 mi/hr, and the diameter of its wheels is about 2 feet. Find the angular speed of the wheels in radians per minute. (1mi = 5280 ft)
  5. Evaluate $\sec(40\degree 10’)$
  6. The point (-8, 15) is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions.
  7. You are 50ft from a building. The angle of elevation to the top of the building is 30 degrees. How tall is that building?
  8. What is the measure of the smallest angle of a 3-4-5 triangle in radians?
  9. A signal buoy bobs up and down with height h of its transmitter that is 5 feet above sea level. The waves vary from 1ft to 9ft, and there are 3.5 seconds between the maximun heights. Write a model that represents the height of the transmitter from sea level, starting at its maximun height, during this period of waves.



  1. $f(x) = \csc\frac x2 + 3$
  2. $f(x) = 2\tan\pi x -2$
  3. $f(\theta) = -\cot 2\theta$
  4. $f(y) = 3\sec(3y-\pi)+1$
  5. For $f(x) = 0.5\sec(2x-\pi)$,
    1. Where are all the asymptotes?
    2. What is the domain?
    3. What is the range?
  6. Is tangent an odd or even function?
  7. Find $x$ for $\csc x = 1$ on $[0,2\pi)$ (no calculator)
  8. Find $x$ for $\cot x = 1$ on $[0,2\pi)$ (no calculator)