ACT Math: Right Triangle Trigonometry (SOH-CAH-TOA) Problems

Below you’ll find seven problems similar to those found on the ACT Math test.  You can solve each of these using what you know about right triangle trigonometry.  Some people call these SOH-CAH-TOA problems.  This is a clever acronym to remember the sine, cosine and tangent ratios.

The answers appear at the bottom of this post.

(Note: If you’d like to try even more of these problems, click here when you’re done.)

1)  In the triangle below, what is the value of cos A?


A.  3/5
B.  3/4
C.  4/5
D,  5/4
E.  5/3

2)  A vertical ramp is constructed for a loading dock.  The ramp is 50 feet long and rises 4 feet vertically at its highest point.  What angle does the ramp make with the ground?  Round your answer to the nearest tenth of a degree.

F.  3.1
G.  4.2
H.  4.6
J.   5.2
K.  8.0

3)  The hypotenuse of the right triangle JKL is 14 feet long.  The cosine of angle J  is 7/10.  About how many feet long is side JK?

A.  7.0
B.  8.4
C.  9.8
D.  12.4
E.  20.0

4)  Jon stands at point A, which is 300 feet from the launch point of a hot air balloon.  The balloon is rising vertically at a rate of 6 feet per second.  What is the angle of elevation from Jon’s position to the balloon 10 seconds after the balloon lifts off the ground?  Round your answer to the nearest tenth of a degree.


F.  8.6
G.  10.4
H.  10.8
J.   11.3
K.  12.0

5)  Consider the right triangle below with side measures as shown, measured in the same units.  For this triangle, (sin A)(sin B)(tan A) is equivalent to:


A.  a/b
B.  a/c
C.  ab/c2
D.  a2/c2
E.  a2/bc

6)  In the right triangle XYZ what is an expression for the secant of angle X ?


F.  \frac{x}{y}
G.  \frac{y}{x}
H.  \frac{\sqrt{x^2+y^2}}{x}
J.  \frac{\sqrt{x^2+y^2}}{y}
K. \frac{y}{\sqrt{x^2+y^2}}

7)  If \cos\theta  =  -\frac{5}{13}  and  \frac{\pi }{2}  <  \theta  <  \pi, then tan\, \theta = ?

A.  – 12/5
B.  -12/13
C.  – 5/12
D.  5/12
E.  12/5

If you’d like more practice on this type of problem, click here.

If you have questions about these problems or anything else to do with the ACT, leave a comment below or send me an email at .

1.  C
2.  H
3.  C
4.  J
5.  D
6.  J
7.  A

(Note: If you need a little bit of review on the more advanced trigonometry problems — like #7 — you may want to read this post.)

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