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.
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 35∘ 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?
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Answer: C — 120sin35∘
The kite’s height is the side opposite the 35∘ angle, and the string is the hypotenuse, so height =120sin35∘.
Question 2Easier
What is the amplitude of the function f(x)=−3cos(5x−π)?
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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≤x<2π,tan(2x)=3. Which of the following is the set of all possible values of x?
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Answer: B — {6π,32π,67π,35π}
Tangent equals 3 when its argument is 3π+kπ, so 2x=3π+kπ and x=6π+2kπ. On 0≤x<2π that gives 6π,32π,67π, and 35π.
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