In right triangle , is a right angle. If and , what is ?

Solution available with step-by-step explanation

SAT Right Triangles & Trigonometry Question 34 | Free SAT Prep | Aniko

00:00

Question 34·Easy·Right Triangles and Trigonometry

0 / 50

In right triangle $\triangle ABC$ , $\angle B$ is a right angle. If $AB = 5$ and $BC = 12$ , what is $\sin A$ ?

For right-triangle trig questions, quickly label the sides relative to the given angle as opposite, adjacent, and hypotenuse, then recall the basic ratios (SOH-CAH-TOA: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent). If one side is missing, use the Pythagorean theorem or recognize common triples like 3-4-5 or 5-12-13 to find it, then plug directly into the appropriate ratio and immediately eliminate any answer greater than 1 for sine or cosine.

Hints

Click me!

Label the sides relative to angle A

First, decide which side is the hypotenuse (it is opposite the right angle) and which side is opposite angle A. Which given side length is directly across from angle A?

Find the missing side

You know two sides of a right triangle (5 and 12). Use the Pythagorean theorem, or recall any common Pythagorean triple that includes 5 and 12, to find the third side.

Use the definition of sine

For angle A, sine is the ratio of the length of the side opposite angle A to the length of the hypotenuse. Once you know all three sides, form this ratio and compare it to the answer choices.

Desmos Guide

Compute the hypotenuse length

In Desmos, type sqrt(5^2 + 12^2) to calculate the length of the hypotenuse $AC$ . Note the numerical value that appears.

Form the sine ratio numerically

Now type 12 / sqrt(5^2 + 12^2) to compute the value of $\sin A$ as opposite over hypotenuse. Observe the decimal result Desmos gives.

Compare with answer choices

Enter each choice as a fraction in Desmos: 5/12, 12/13, 5/13, and 13/12. Compare their decimal values to the value from Step 2; the fraction whose decimal matches is the correct answer.

Step-by-step Explanation

Identify the sides of the triangle

Since angle B is a right angle, side $AC$ (across from B) is the hypotenuse. The legs are $AB = 5$ and $BC = 12$ .

From angle A:

The side opposite angle A is $BC$ , which has length 12.
The side adjacent to angle A is $AB$ , which has length 5.
The hypotenuse is $AC$ , whose length we do not yet know.

Find the hypotenuse using the Pythagorean theorem

Use the Pythagorean theorem for right triangle $\triangle ABC$ with right angle at B:

AC^2 = AB^2 + BC^2

Substitute the given values:

AC^2 = 5^2 + 12^2 = 25 + 144 = 169

So $AC = 13$ .

Recall the definition of sine for an acute angle

In a right triangle, for an acute angle (like angle A),

\sin(\text{angle}) = \dfrac{\text{opposite side}}{\text{hypotenuse}}

So,

\sin A = \dfrac{\text{length of side opposite A}}{\text{length of hypotenuse}}

From Step 1, the opposite side to angle A is $BC$ ; from Step 2, the hypotenuse is $AC$ .

Compute $\sin A$ and match it to the choices

We now substitute the side lengths into the sine ratio:

\sin A = \dfrac{BC}{AC} = \dfrac{12}{13}

Thus $\sin A = \dfrac{12}{13}$ , which corresponds to answer choice B ( $\dfrac{12}{13}$ ).

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation