Mathfolis

Right Triangles & Trigonometry

Unit 4 · Geometry & Trigonometry

What the SAT asks here

Right-triangle problems on the SAT use three tools: the Pythagorean theorem, the side ratios of the 45-45-90 and 30-60-90 special triangles, and SOH-CAH-TOA for sine, cosine, and tangent. The complementary-angle identity (sin θ = cos(90° − θ)) shows up often. Expect 1–2 problems per test.

Pythagorean theorem

Legs a, b; hypotenuse c
a2+b2=c2a^2 + b^2 = c^2

Special right triangles

45-45-90 (sides opposite the listed angles)
1:1:21 : 1 : \sqrt{2}
30-60-90 (sides opposite the listed angles)
1:3:21 : \sqrt{3} : 2

SOH-CAH-TOA

sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}
tan⁡θ=oppositeadjacent\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

Identities to know

Complementary angles
sin⁡θ=cos⁡(90∘−θ)\sin \theta = \cos(90^\circ - \theta)
Pythagorean identity
sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
SAT Tip: Pythagorean triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) come up constantly. Recognise them on sight to skip the arithmetic.
Caution: The 'opposite' side is across from the angle θ — not next to it. Mixing up opposite and adjacent flips sin and cos.
Type 1

Solve a right triangle

Apply the Pythagorean theorem or a basic trig ratio to find a missing side or angle.

Example 1
In a right triangle, the legs are 5 and 12. Find the hypotenuse.

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Type 2

Special right triangles

Match the given side to the right position in the 45-45-90 (1 : 1 : √2) or 30-60-90 (1 : √3 : 2) ratio, then scale.

Example 2
A 30-60-90 triangle has hypotenuse 14. Find the length of the side opposite the 30° angle.

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Type 3

Trig ratio or identity

Use SOH-CAH-TOA, the complementary-angle rule, or sin²θ + cos²θ = 1 to relate angle measures to side ratios.

Example 3
If sin θ = 3/5 and θ is acute, find cos θ.

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