Practice Trigonometry for CAT

If you’re studying for the Common Admission Test (CAT), then you should know that trigonometry makes an appearance in the Quantitative section. This article will provide you a set of practice problems involving trigonometric concepts similar to those found on the CAT. Detailed solutions are given for you to check your work afterward.

Trigonometry on a circle

Trigonometry on a circle (from a 1902 textbook; public domain)

Short Review of Trigonometry

First, let’s review the fundamentals.

Triangle Trig

Suppose that triangle ABC is right, and assume that ∠ C is the right angle. To find the sine, cosine, and tangent of ∠ A, just remember: SOHCAHTOA!

right triangle with edges and angles labeled

SOHCAHTOA - trig ratios

The tangent can also be defined in terms of sine and cosine — as can the three other trigonometric ratios. Let’s call θ = ∠ A.

four other trig ratios

The Unit Circle

You should also be familiar with the Unit Circle. Each point (x, y) on the circle gives the value of cosine and sine of the corresponding angle.

For example, look for the angle 2π/3 (radians). Based on the unit circle diagram, you know:

  • cos 120° = -1/2.
  • sin 120° = √ 3 /2.
Unit circle - trigonometric identities

The Unit Circle, which displays cosine and sine values of common angles.

Trigonometric Functions

The six basic trig ratios can be extended to functions defined on all (or most) real numbers by interpreting arbitrary angles on the unit circle. The key is that each trig function becomes periodic, which means that values repeat in equal intervals.

It’s important to know the periods, domains, and ranges of each function.

Period, domain, and range for all six trig functions

Period, domain, and range for all six trig functions.

The Pythagorean Identities

Perhaps the most famous and useful equation in all of mathematics is the Pythagorean Theorem. (For this and other essentials about triangles, check out: Triangle Properties to Know for the CAT).

When applied to the unit circle, we get a trio of useful trigonometric identities:

Other Useful Identities

Of course there are a myriad of other trigonometric identities, including sums or differences of angles, half and double angle formulas, products-to-sums, and sums-to-products, to name a few.

For a summary list, check out this table of trigonometric identities.

CAT Trigonometry Practice Problems

Now let’s test our knowledge and skills!

  1. You are standing on the corner of a square whose side length is 25 feet. Standing on the opposite corner from you is a tall tree. The angle of elevation from your position to the top of the tree is exactly 60°. How tall is the tree?

     

    A. 25√ 2 

    B. 25√ 3 

    C. 25√ 6 

    D. 50√ 3 

  2. Tweedledee and Tweedledum are 100 m from each other. Between them there is a tower. Tweedledee notes that the top of the tower is at x°, while Tweedledum records that the top of the tower is at y°. Which expression below correctly computes the height of the tower?

     

    A. 100 tan x° tan y° / (tan x° + tan y°)

    B. 100 (tan x° + tan y°) / (tan x° – tan y°)

    C. 100 (tan x° + tan y°) / (tan x° tan y°)

    D. 100 tan x° tan y° / (tan x° – tan y°)

  3. What is the maximum value of 8 sin θ + 6 cos θ?

     

    A. 9.5

    B. 10

    C. 10.3

    D. 10.8

  4. Suppose the angle of elevation of the top of a flag pole changes from x° to 45° as you walk 15 m toward it. Assuming that x < 45, find the height of the flag pole in terms of x.

     

    A. 30(1 + tan x°) / tan x°

    B. 45 tan x° / (1 + tan x°)

    C. 15 tan x° / (1 + tan x°)

    D. 15 tan x° / (1 – tan x°)

Solutions

  1. C.

    First find the distance of the diagonal d along the ground from corner to corner. Using Pythagorean theorem with sides 25 and 25, we get:

    252 + 252 = d2

    2 × 252 = d2

    d = 25√ 2 .

    Then to obtain the height h of the tree, use the tangent ratio with angle 60°.

    tan 60° = x / (25√ 2 )

     3  = x / (25√ 2 )

    x = 25√ 2  × √ 3  = 25√ 6 

  2. A.

    First let d represent the distance from Tweedledee to the tower. Then the distance from Tweedledum to the tower must be 100 – d. Next, let h be the height of the tower. So, there are two right triangles, one determined by Tweedledee and the other by Tweedledum, sharing the same height. Finally, use trigonometry to write two equations:

    problem2_trig_solution_partA

    Next, solve the first equation for d:

    d tan x° = h   →   d = h / tan x°

    Now plug this result into the second equation and isolate h.

    problem2_trig_solution_partB

  3. B.

    Here, the clue is that 6 and 8 form part of a Pythagorean Triple: 6-8-10.

    Let f = 8 sin θ + 6 cos θ, and divide both sides by the constant 10 to get:

    f/10 = (8/10) sin θ + (6/10) cos θ

    Because 10 is constant, it’s equivalent to find a maximum value for f/10.

    Next, we can assume that θ is a first-quadrant angle (otherwise, either its sine value or cosine value would be negative). So construct a right triangle with angle θ.

    Now the trick is to compare this triangle to the one whose sides are in ratio 6 : 8 : 10. In fact, in a triangle whose adjacent side is 8 and opposite side is 6, then the hypotenuse would be 10. So, calling the angle of this new triangle φ, we can identify:

    Then, sin φ = 6/10 and cos φ = 8/10.

    Fortunately, these ratios already show up in our equation. (That’s why I picked the 6-8-10 triangle to compare to!) So we may substitute for the trig ratios:

    f/10 = cos φ sin θ + sin φ cos θ

    Next, using the sum of angles formula for sine, we could write:

    f/10 = sin(φ + θ)

    What is the maximum value of this function? Remember, the range of sin x is from -1 up to 1. So the maximum occurs when sin(φ + θ) = 1. This will happen as long as φ + θ = 90°. In other words, the unknown angle θ must be complementary to the angle of a 6-8-10 triangle. That makes our unknown triangle into a 6-8-10 triangle as well!

    But the bottom line is that we can find such an angle, and so the maximum sine value (1) will be obtained. Finally, use f/10 = 1 (maximum) to find that f = 10.

  4. D.

    Let h be the height of the flag pole. When you reach the point at which the angle of elevation is exactly 45°, then you are at the base of an isosceles right triangle with height h. Therefore, at that point, you are h meters from the base of the flag pole. Thus your original position was 15 + h meters from the pole.

    This information allows us to set up a trigonometric equation:

    problem4_solution

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