Trigonometry Practice Problems

Trigonometry deals with the measurements of triangles both angles and sides. Many a real life situations are modeled using trigonometric functions. Trigonometric functions are transcendental in nature which are a study of importance in calculus. A good skill in solving trigonometric problems is therefore essential to acquire a high level of proficiency in Calculus.

A practice test is given here. You can solve the test problems first on your own and verify with the answers given in the end. Hints to solve or important steps for each problem are also given.

Trigonometry Practice test

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1.  Rewrite the each degree measure a radian measure as a multiple of π.
     (a) 30º (b) 315º (c) -20º (d) 720º
2. Rewrite each radian measure as a degree measure
    (a) $\frac{7\pi }{6}$  (b) $-\frac{11\pi }{30}$  (c)$\frac{3\pi }{2}$ (d) $\frac{2\pi }{9}$
3. A sprinkler system can spray water over a distance of 40 meters and can rotate through an angle or 135º. Find the maximum area that can
    be irrigated by the  sprinkler.
4. If sin θ = `3/8` where 0 ≤ θ ≤ $\frac{\pi }{2}$, then evaluate the exact values of the other five trigonometric functions.
5. If sin θ = $\frac{2\sqrt{13}}{13}$, find the value of sec (90 - θ) where θ is an acute angle.

6. Find the value of x in the adjoining

7. Bill was driving on a straight and flat road. He observed the peak of a mountain directly in front of him.  He estimated the angle of
    elevation of the peak as 4.5 º. After driving another 15 miles ahead in the same road, he again estimated the angle of elevation of
    the peak to be 10 º. Find the approximate height of the mountain using trigonometric relationships.

8. Use the trigonometric identities to transform the left side of the equation to the right side.
    $\frac{sin \theta }{cos \theta }+\frac{cos\theta }{sin\theta }= csc\theta .sec\theta $

9. The displacement from equilibrium of an oscillating weight suspended by a spring is given by
    y(t) = 2 cos 8t, where y(t) is the displacement in cm for time 't' seconds. Find the displacements when t = 0, `1/4` and `1/2`.
    Also find the period and frequency of the motion.

10. While running on a circular track, the angle of incline θ is the acute angle which the runner's body makes with the vertical. The relationship
      between θ, the velocity ' v ' of the runner and the radius ' r ' of the track is given by
      $tan\theta =\frac{v^{2}}{gr}$, where ' g ' is the acceleration due to gravity and is equal to 9.8 meters/second2.
      If the radius of the track is 16 meters,
  1.       Find the runner's velocity if the angle of incline is 13 º.
  2.       What is the angle of incline if the runner's velocity is read as 7 meters /second ?

11.  An air plane flying horizontally 750 meters above the ground is first observed at an angle of elevation of 60º. If the angle is elevation is
      is observed to be 30º after 5 seconds, find the speed of the air plane  rounded to a Km per hour.
12.  A jet plane leaves city A and is heading toward city at a bearing of 120º. If the distance between the two cities are 2600 KM,
      1. How far north and how far west is city  A relative to city B. Round the answers to the nearest KM.
       2. If the jet is to return directly from city B to city A, at what bearing should it travel?
       Hint: For air navigation a bearings are measured clockwise from north.

Trigonometry Practice  test -  answers and hints

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1.  (a) $\frac{\pi }{6}$  (b) $\frac{7\pi }{4}$  (c) -$\frac{\pi }{9}$  (d) 4π
     Hint: Conversion factor for multiplication is $\frac{\pi }{180}$  and retain the same sign.

2.  (a) 210º  (b) -66º   (c) 270º (d) 40º
     The conversion factor for multiplication is $\frac{180}{\pi }$

3.  600 π m2 or ≈ 1884.96 m2.  (Use the formula for the area of the sector = `1/2`.r2θ   where θ is expressed in radians.

4. Cos θ = $\frac{\sqrt{55}}{8}$   tan θ = $\frac{3\sqrt{55}}{55}$   Sec θ = $\frac{8\sqrt{55}}{55}$
    Csc θ = `8/3`  Cot θ = $\frac{\sqrt{55}}{3}$ 
    Hint make a sketch of a right triangle as shown below. Find the value of X using Pythagorean theorem.

5.  $\frac{\sqrt{13}}{2}$  Use the cofunction identity to find the value of Cos (90 - θ).

6.  30√3.   Use the properties of the special 30-60-90 right triangle.

7.  11,258.12 ft.  Sketch the situation as shown and evaluate h eliminating x.

8.  Simplify the left side by taking the LCD.

9.  y(0) = 2, y(`1/4`) = -0.8323  y(`1/2) = -1.3073 
     Period of the motion = $\frac{\pi }{4}$ seconds
     Frequency of the motion = $\frac{4}{\pi }$ cm/second.
     Hint; Use the radian mode in the calculator.

10.  (1) 6.017 meter/second   (2) 17.35º
       Solve the equation for the required variable.

11.  624 Km/hour.
       The sketch for the situation is given below. Considering the two right triangles OCA and ODB, solve for 'd' which is the distance traveled
       by the air plane in 5 minutes.  Use this value to determine the speed of the air plane.

  1. A is at a distance of 1300 KM North and 2252 KM west of city B.
  2. The bearing on the return journey is 240 º.

      The sketch for the situation is given with the bearing marked. Bearings are measured in clockwise for air navigation.

      The North and West distances are represented by the lengths BC and CA respectively.