Right-triangle trigonometry and trig functions — ACT practice questions

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Practice 61 Right-triangle trigonometry and trig functions questions in the app

What this topic is

SOH-CAH-TOA is the starting point for Right-triangle trigonometry and trig functions: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

Word problems about ramps, ladders or angles of elevation usually ask for an expression, such as a length times the sine of an angle, rather than a decimal. Beyond triangles, the ACT covers graphs of trig functions, including amplitude, which is the absolute value of the number multiplying sine or cosine, and period.

Choosing cosine where sine belongs is the usual mistake, and the choices are built to catch it.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

A kite string is attached to the ground and pulled taut so that it makes a $3 5^{\circ}$ angle with the ground. If the string is 120 feet long, which of the following expressions gives the kite’s height, in feet, above the ground?

120 \cos 3 5^{\circ}

✗ Not this one

120 \tan 3 5^{\circ}

✗ Not this one

120 \sin 3 5^{\circ}

✓ Correct

\frac{120}{\tan 3 5^{\circ}}

✗ Not this one

Show answer

Answer: C — $120 \sin 3 5^{\circ}$

The kite’s height is the side opposite the $3 5^{\circ}$ angle, and the string is the hypotenuse, so height $= 120 \sin 3 5^{\circ}$ .

Question 2Easier

What is the amplitude of the function $f (x) = - 3 \cos (5 x - π)$ ?

- 3

✗ Not this one

\frac{1}{3}

✗ Not this one3✓ Correct5✗ Not this one

Show answer

Answer: C — 3

Amplitude is $∣ A ∣ = ∣ - 3 ∣ = 3$ . The sign of $A$ reflects the graph but does not change amplitude; 5 is the frequency coefficient.

Question 3Mid

For $0 \leq x < 2 π$ , $\tan (2 x) = \sqrt{3}$ . Which of the following is the set of all possible values of $x$ ?

{\frac{π}{6}, \frac{7 π}{6}}

✗ Not this one

{\frac{π}{6}, \frac{2 π}{3}, \frac{7 π}{6}, \frac{5 π}{3}}

✓ Correct

{\frac{π}{3}, \frac{4 π}{3}}

✗ Not this one

{\frac{π}{6}, \frac{π}{3}, \frac{7 π}{6}, \frac{4 π}{3}}

✗ Not this one

Show answer

Answer: B — ${\frac{π}{6}, \frac{2 π}{3}, \frac{7 π}{6}, \frac{5 π}{3}}$

Tangent equals $\sqrt{3}$ when its argument is $\frac{π}{3} + k π$ , so $2 x = \frac{π}{3} + k π$ and $x = \frac{π}{6} + \frac{k π}{2}$ . On $0 \leq x < 2 π$ that gives $\frac{π}{6}, \frac{2 π}{3}, \frac{7 π}{6},$ and $\frac{5 π}{3}$ .

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 61 Right-triangle trigonometry and trig functions questions in the app

Right-triangle trigonometry and trig functions — ACT practice questions

What this topic is

Sample questions

More ACT topics

Mathematics