If sin 25^ = cos x, find x. [1 credit]

NYSED Regents (style)-style practice: Part I (multiple choice). In right triangle ABC with the right angle at C, A = 35^ and the hypotenuse AB = 20. Which expression gives the length of side BC (opposite A)? (1) 20sin 35^ (2) 20cos 35^ (3) (20)/(sin 35^) (4) 20tan 35^

The correct answer is (1). Side BC is opposite the 35^ angle, and AB = 20 is the hypotenuse. Sine relates opposite to hypotenuse: sin 35^ = (BC)/(20), so BC = 20sin 35^. Choice (2) uses cosine, which is for the adjacent side; choice (4) uses tangent, which needs the adjacent side, not the hypotenuse.

New YorkMathsSyllabus dot point

How do the sine, cosine, and tangent ratios let you find missing sides and angles in a right triangle?

Define the sine, cosine, and tangent ratios in a right triangle (SOHCAHTOA), use them with inverse trig functions to find missing sides and angles, apply the relationship between the sine and cosine of complementary angles, and solve angle-of-elevation and depression problems.

A NY Regents Geometry answer on right triangle trigonometry: the sine, cosine, and tangent ratios, inverse trig to find an angle, the complementary sine-cosine relationship, and angle of elevation and depression problems.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

In a right triangle, the trig ratios relate an acute angle to the sides: SOHCAHTOA means $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$ , $\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ , and $\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$ . To find a side, set up the ratio that uses the side you want and a known side, then solve. To find an angle, use the inverse ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ ) of the appropriate ratio. The sine of an angle equals the cosine of its complement: $\sin\theta = \cos(90^\circ - \theta)$ . An angle of elevation looks up and an angle of depression looks down, measured from the horizontal.

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What this topic is asking
The three ratios: SOHCAHTOA
Finding an angle with inverse trig
Complementary angles and applications
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What this topic is asking

The Regents Geometry exam (the Similarity, Right Triangles, and Trigonometry, G-SRT, cluster) wants you to use the three trig ratios, sine, cosine, tangent, to find missing sides and angles in a right triangle, to use inverse trig functions to recover an angle, to apply the complementary relationship between sine and cosine, and to solve angle-of-elevation and depression problems. Right-triangle trig is a reliable source of credits in every part of the exam.

The three ratios: SOHCAHTOA

Each acute angle of a right triangle has three trig ratios built from the opposite side, the adjacent side, and the hypotenuse (the side opposite the right angle).

To find a missing side, choose the ratio that involves the side you want and one you know, then solve. The choice of opposite versus adjacent is always relative to the angle you are using.

Finding an angle with inverse trig

When you know two sides and want the angle, apply the inverse trig function. If $\tan\theta = \frac{5}{8}$ , then $\theta = \tan^{-1}\!\left(\frac{5}{8}\right) \approx 32^\circ$ . Choose the inverse that matches the two sides you have: opposite and hypotenuse use $\sin^{-1}$ , adjacent and hypotenuse use $\cos^{-1}$ , opposite and adjacent use $\tan^{-1}$ .

Finding a side and an angle

A right triangle has a hypotenuse of 26 and one leg of 10, with the right angle between the two legs. (a) Find the other leg. (b) Find the angle opposite the leg of length 10, to the nearest degree.

step 1 Find the missing leg with the Pythagorean theorem

$26^2 = 10^2 + b^2$ , so $676 = 100 + b^2$ , giving $b^2 = 576$ and $b = 24$ .

step 2 Identify the sides relative to the target angle

For the angle $\theta$ opposite the leg of length 10: opposite $= 10$ , hypotenuse $= 26$ .

step 3 Set up the inverse sine

$\sin\theta = \frac{10}{26}$ , so $\theta = \sin^{-1}\!\left(\frac{10}{26}\right)$ .

step 4 Evaluate and round

$\theta = \sin^{-1}(0.3846) \approx 23^\circ$ . The angle opposite the shorter leg is about 23 degrees.

Complementary angles and applications

In a right triangle the two acute angles are complementary (sum to 90 degrees), and this links sine and cosine: $\sin\theta = \cos(90^\circ - \theta)$ . So $\sin 40^\circ = \cos 50^\circ$ , a relationship the Regents tests directly.

An angle of elevation is measured upward from the horizontal to a higher object; an angle of depression is measured downward to a lower object. A clarifying point worth stressing is that the angle of elevation from a lower point equals the angle of depression from the higher point, because they are alternate interior angles between parallel horizontal lines. In a word problem, draw the right triangle, label the angle from the horizontal, identify which sides are opposite and adjacent, and pick the ratio accordingly. Choosing the wrong ratio is the most common error, so always label opposite and adjacent before writing the equation.

Try this

Q1. In a right triangle, the side adjacent to a 50-degree angle is 8 and the hypotenuse is unknown. Which ratio finds the hypotenuse? [1 credit]

Cue. Cosine relates adjacent and hypotenuse: $\cos 50^\circ = \frac{8}{h}$ .

Q2. If $\sin 25^\circ = \cos x$ , find $x$ . [1 credit]

Cue. Sine equals cosine of the complement: $x = 90 - 25 = 65$ degrees.

Exam-style practice questions

Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Regents (style)2 marksPart I (multiple choice). In right triangle

ABC

with the right angle at

C

\angle A = 35^\circ

and the hypotenuse

AB = 20

. Which expression gives the length of side

BC

(opposite

\angle A

)? (1)

20\sin 35^\circ

(2)

20\cos 35^\circ

(3)

\frac{20}{\sin 35^\circ}

(4)

20\tan 35^\circ

Show worked answer →

The correct answer is (1).

Side $BC$ is opposite the $35^\circ$ angle, and $AB = 20$ is the hypotenuse. Sine relates opposite to hypotenuse: $\sin 35^\circ = \frac{BC}{20}$ , so $BC = 20\sin 35^\circ$ . Choice (2) uses cosine, which is for the adjacent side; choice (4) uses tangent, which needs the adjacent side, not the hypotenuse.

Regents (style)4 marksPart III (constructed response). A ladder leans against a wall, reaching 12 feet up the wall, with its base 5 feet from the wall. (a) Find the angle the ladder makes with the ground, to the nearest degree. (b) Find the length of the ladder.

Show worked answer →

A 4-credit question: credit for the correct trig setup, the angle, and the length.

(a) The height (12, opposite) and base distance (5, adjacent) give the tangent: $\tan\theta = \frac{12}{5} = 2.4$ , so $\theta = \tan^{-1}(2.4) \approx 67^\circ$ .
(b) The ladder is the hypotenuse: by the Pythagorean theorem $L = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13$ feet. Using the wrong ratio (for example sine with the base) or forgetting to round the angle as requested costs a credit.

Related dot points

Sources & how we know this

Regents Examination in Geometry — NYSED (2024)
New York State Next Generation Mathematics Learning Standards — NYSED (2017)