Study on the flyACT

Study on the fly›ACT›Mathematics › Right-triangle trigonometry and trig functions

MathematicsFree

Right-triangle trigonometry and trig functions — ACT practice questions

Right-triangle trigonometry and trig functions on the ACT Mathematics test covers sine, cosine, and tangent in geometric settings and the graphs and equations of trigonometric functions. A student must pick the correct ratio to find a missing length, identify amplitude, and find every solution on an interval such as 0≤ x<2π. Items often show a taut string or similar right triangle and ask which expression gives a height, or they ask for amplitude or every admissible value of x in an interval. Traps include the wrong ratio, mixing amplitude with period, and omitted solutions. Choices are expressions, single values, or listed sets of angles.

  • Section: Mathematics 45 questions on the paper
  • 56 practice questions in the app

Sample questions

Question 1

A kite string is attached to the ground and pulled taut so that it makes a 35∘35^\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?

  1. A120cos⁡35∘120\cos 35^\circ
  2. B120tan⁡35∘120\tan 35^\circ
  3. C120sin⁡35∘120\sin 35^\circ
  4. D120tan⁡35∘\dfrac{120}{\tan 35^\circ}
Show answer

Answer: C — 120sin⁡35∘120\sin 35^\circ

The kite's height is the side opposite the 35∘35^\circ angle, and the string is the hypotenuse, so height =120sin⁡35∘= 120\sin 35^\circ.

Question 2

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

  1. A-3
  2. B13\dfrac{1}{3}
  3. C3
  4. D5
Show answer

Answer: C — 3

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

Question 3

For 0≤x<2π0\le x<2\pi, tan⁡(2x)=3\tan(2x)=\sqrt{3}. Which of the following is the set of all possible values of xx?

  1. A{π6,7π6}\left\{\dfrac{\pi}{6},\dfrac{7\pi}{6}\right\}
  2. B{π6,2π3,7π6,5π3}\left\{\dfrac{\pi}{6},\dfrac{2\pi}{3},\dfrac{7\pi}{6},\dfrac{5\pi}{3}\right\}
  3. C{π3,4π3}\left\{\dfrac{\pi}{3},\dfrac{4\pi}{3}\right\}
  4. D{π6,π3,7π6,4π3}\left\{\dfrac{\pi}{6},\dfrac{\pi}{3},\dfrac{7\pi}{6},\dfrac{4\pi}{3}\right\}
Show answer

Answer: B — {π6,2π3,7π6,5π3}\left\{\dfrac{\pi}{6},\dfrac{2\pi}{3},\dfrac{7\pi}{6},\dfrac{5\pi}{3}\right\}

Tangent equals 3\sqrt3 when its argument is π3+kπ\dfrac{\pi}{3}+k\pi, so 2x=π3+kπ2x=\dfrac{\pi}{3}+k\pi and x=π6+kπ2x=\dfrac{\pi}{6}+\dfrac{k\pi}{2}. On 0≤x<2π0\le x<2\pi that gives π6,2π3,7π6,\dfrac{\pi}{6},\dfrac{2\pi}{3},\dfrac{7\pi}{6}, and 5π3\dfrac{5\pi}{3}.

Practice 56 Right-triangle trigonometry and trig functions questions in the app

More ACT topics

Mathematics

Everything on the ACT

Study on the fly is an independent study tool and is not affiliated with, sponsored by, or endorsed by any testing organization, school, or government body.

ACT® is a registered trademark of ACT Education Corp., which is not affiliated with, and does not endorse, this product.

All practice questions and passages are original works created for Study on the fly and are not reproduced from any official test.