In a right triangle, the measures of the two acute angles are x^ and (2x - 30)^. It is also true that (x^) = ((2x-30)^). What is the value of x?

SAT Right triangles and trigonometry Practice Question 5 (Hard)

Question bank / Right triangles and trigonometry

Math · Geometry and TrigonometryHard

In a right triangle, the measures of the two acute angles are $x^\circ$ and $(2x - 30)^\circ$ . It is also true that $\sin(x^\circ) = \cos((2x-30)^\circ)$ . What is the value of $x$ ?

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Answer: 40

The two acute angles of a right triangle are complementary, so $x + (2x - 30) = 90$ . This gives $3x = 120$ , so $x = 40$ . This is consistent with the given equation, because the sine of an angle equals the cosine of its complement.

Setting $x = 2x - 30$ gives $x = 30$ , but $\sin a = \cos b$ means the angles are complementary, not equal.

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