SAT Saturday: Algebra & Linear Models

In right triangle ABC, angle B is a right angle. If cos(A) = 5/13, what is the value of sin(C)?

• A) 5/13
• B) 12/13
• C) 13/12
• D) 12/5

Correct Answer: A)

The Concept:

This problem highlights one of the SAT’s favorite trigonometric identities:

sin (x°) = cos (90° – x°)

The Breakdown:

1. In any right triangle, the two acute angles always sum to 90° (they are complementary). Because angle B is the  right angle, angles A and C must add up to 90°.
2. The sine of one acute angle in a right triangle is always exactly equal to the cosine of the other acute angle. Why? Because the “opposite” side for angle C is the exact same physical side as the “adjacent” side for angle A.
3. Therefore:

sin (C) = cos (A) = 5/13

No complex drawings or side-length calculations required!

Triangle JKL is similar to triangle XYZ, where vertices J, K, and L correspond to vertices X, Y, and Z, respectively. The measure of angle K is 90°, the length of side JK = 15, and the length of side KL = 8. What is the value of tan(Z)?

Correct Answer: 15/8 (or 1.875)

The Concept:

This problem tests two concepts at once: right-triangle ratios (tan Θ = opposite/adjacet) and similar triangle properties.

Step-by-Step Breakdown:

1. Apply the Similar Triangles Rule: Similar triangles have proportional side lengths, but their corresponding angles are identical. Because the angles are identical, their trigonometric ratios (sin, cos, tan) are also identical.
2. Since vertex L corresponds to vertex Z, we know that tan Z = tan L. We can do all of our work directly using triangle JKL!
3. Apply SOH-CAH-TOA: Angle K is the 90° right angle, which means sides JK and KL are the legs. For angle L:
   • The side opposite to L is JK = 15.
   • The side adjacent to L is KL = 8.

tan (L) = opposite/adjacent = 15/8

Students can grid in the fraction 15/8 or the decimal equivalent 1.875.