Before you read

Answer first — the lesson below explains it.

A right-triangular ramp $A BC$ has $C = 90°$ , hypotenuse $A B = 10$ ft, and vertical rise $BC = 6$ ft. What is $sin A$ ?

A
$\frac{3}{5}$
B
$\frac{4}{5}$
C
$\frac{3}{4}$
D
$\frac{5}{3}$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A wheelchair ramp makes a $30°$ angle with the ground and is 20 ft long. How high is the ramp at its top end? ( $sin 30° = 0.5$ )

A
10 ft
B
15 ft
C
$103$ ft
D
20 ft

Check your answer

Answer:

Theory

SOH CAH TOA

In a right triangle, for an acute angle $\theta$ :

$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \qquad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \qquad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$

SOH CAH TOA: sin = 3/5, cos = 4/5, tan = 3/4

Memory aid: SOH-CAH-TOA

Also: $\tan\theta = \frac{\sin\theta}{\cos\theta}$

Example 1

In a right triangle, the side opposite $\theta$ is 5 and the hypotenuse is 13. Find $\sin\theta$ , $\cos\theta$ , $\tan\theta$ .

Adjacent $= \sqrt{13^2 - 5^2} = 12$

$\sin\theta = \frac{5}{13}$ , $\cos\theta = \frac{12}{13}$ , $\tan\theta = \frac{5}{12}$

\sin\theta = \frac{5}{13}

\cos\theta = \frac{12}{13}

\tan\theta = \frac{5}{12}

Quick check

You just read it. Can you apply it?

A ladder leans against a building. If $sin x = 0.6$ (where $x$ is the angle with the ground) and $x$ is acute, what is $cos x$ ?

A
0.4
B
0.5
C
0.8
D
1.0

Check your answer

Answer:

$sin^{2} x + cos^{2} x = 1$ . $cos^{2} x = 1 - 0.36 = 0.64$ . $cos x = 0.8$ .

Strategy

SAT Pro Tip

The SAT often asks: 'In a right triangle, $\sin A = \frac{3}{5}$ . What is $\cos A$ ?' Draw the triangle: opposite = 3, hypotenuse = 5, so adjacent = 4. $\cos A = \frac{4}{5}$ . Recognize the $3, 4, 5$ triple!

After reading

Same question as before. What do you say now?

A right-triangular ramp $A BC$ has $C = 90°$ , hypotenuse $A B = 10$ ft, and vertical rise $BC = 6$ ft. What is $sin A$ ?

A
$\frac{3}{5}$
B
$\frac{4}{5}$
C
$\frac{3}{4}$
D
$\frac{5}{3}$

Check your answer

Answer:

$sin A = \frac{opposite}{hypotenuse} = \frac{BC}{A B} = \frac{6}{10} = \frac{3}{5}$ .

After reading

Same question as before. What do you say now?

A wheelchair ramp makes a $30°$ angle with the ground and is 20 ft long. How high is the ramp at its top end? ( $sin 30° = 0.5$ )

A
10 ft
B
15 ft
C
$103$ ft
D
20 ft

Check your answer

Answer:

$sin 30° = \frac{h}{20}$ . $h = 20 \times 0.5 = 10$ ft.

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