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Trigonometry — ISEE Upper practice questions

Trigonometry on the ISEE Upper Mathematics Achievement section covers right-triangle sine, cosine, and tangent. A student must identify the opposite side, adjacent side, and hypotenuse relative to a named acute angle and select the expression that equals a missing length. Questions typically present a right triangle with a right angle at a labeled vertex, give one side length and one angle measure, and ask which expression equals another side. Answer choices are trigonometric expressions rather than decimal lengths. Common traps include swapping sine with cosine, using the complementary angle, or treating a given leg as the hypotenuse.

  • Section: Mathematics Achievement 47 questions on the paper
  • 130 practice questions in the app

Sample questions

Question 1

A right triangle NOPNOP has a right angle at OO. Side OP‾\overline{OP} measures 16 centimeters, and ∠PNO\angle PNO measures 42∘42^\circ. Which expression is equal to the length of NP‾\overline{NP}?

  1. A16sin⁡42∘\dfrac{16}{\sin 42^\circ}
  2. Bsin⁡42∘16\dfrac{\sin 42^\circ}{16}
  3. C16sin⁡42∘16\sin 42^\circ
  4. D16cos⁡42∘\dfrac{16}{\cos 42^\circ}
Show answer

Answer: A — 16sin⁡42∘\dfrac{16}{\sin 42^\circ}

Side OPOP is opposite the 42∘42^\circ angle and NPNP is the hypotenuse, so sin⁡42∘=16NP\sin 42^\circ=\dfrac{16}{NP}. Solving for NPNP gives 16sin⁡42∘\dfrac{16}{\sin 42^\circ}.

Question 2

Right triangle GHJGHJ has a right angle at HH. The length of GJ‾\overline{GJ} is 14 meters, and the measure of ∠JGH\angle JGH is 19∘19^\circ. Which expression is equal to the length, in meters, of GH‾\overline{GH}?

  1. A14sin⁡19∘14\sin 19^\circ
  2. B14cos⁡19∘14\cos 19^\circ
  3. C14cos⁡19∘\dfrac{14}{\cos 19^\circ}
  4. Dcos⁡19∘14\dfrac{\cos 19^\circ}{14}
Show answer

Answer: B — 14cos⁡19∘14\cos 19^\circ

Side GHGH is adjacent to the 19∘19^\circ angle and GJGJ is the hypotenuse, so cos⁡19∘=GH14\cos 19^\circ=\dfrac{GH}{14}. Therefore GH=14cos⁡19∘GH=14\cos 19^\circ.

Question 3

In right triangle RSTRST, ∠RST\angle RST is a right angle, ∠SRT\angle SRT measures 27∘27^\circ, and the length of RT‾\overline{RT} is 13 feet. Which of the expressions in the answer choices is equal to the length of ST‾\overline{ST}?

  1. Asin⁡27∘13\dfrac{\sin 27^\circ}{13}
  2. B13sin⁡27∘\dfrac{13}{\sin 27^\circ}
  3. C13cos⁡27∘13\cos 27^\circ
  4. D13sin⁡27∘13\sin 27^\circ
Show answer

Answer: D — 13sin⁡27∘13\sin 27^\circ

Side STST is opposite the 27∘27^\circ angle and RTRT is the hypotenuse, so sin⁡27∘=ST13\sin 27^\circ=\dfrac{ST}{13}. Therefore ST=13sin⁡27∘ST=13\sin 27^\circ.

Practice 130 Trigonometry questions in the app

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