SAT Right Triangle Trigonometry: SOHCAHTOA, Special Triangles & Practice

Right-triangle trigonometry is part of the SAT Geometry and Trigonometry domain.

The core ratios:

[ \sin\theta=\frac{\text{opposite}}{\text{hypotenuse}} ]

[ \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} ]

[ \tan\theta=\frac{\text{opposite}}{\text{adjacent}} ]

Memory device:

SOH CAH TOA

Identify sides relative to the angle

“Opposite” and “adjacent” depend on which acute angle you are using.

The hypotenuse:

• is opposite the 90° angle;
• is always the longest side.

Example

Right triangle:

• angle θ
• opposite = 6
• adjacent = 8
• hypotenuse = 10

[ \sin\theta=6/10=3/5 ]

[ \cos\theta=8/10=4/5 ]

[ \tan\theta=6/8=3/4 ]

Right-triangle relationships like these often connect to the quadratic equations guide, especially when a side length is unknown.

Special triangle: 45-45-90

Side ratio:

[ 1:1:\sqrt2 ]

If each leg = x:

hypotenuse:

[ x\sqrt2 ]

Special triangle: 30-60-90

Opposite 30°:

[ x ]

Opposite 60°:

[ x\sqrt3 ]

Hypotenuse:

[ 2x ]

Simplifying ratios like these sometimes calls on the same skills covered in our polynomials and factoring guide.

Similar triangles

If two triangles are similar, corresponding side ratios are equal.

Trigonometric ratios stay constant for the same acute angle.

This connects trig with similarity.

Complementary angles

In a right triangle:

[ \sin\theta=\cos(90^\circ-\theta) ]

Because the opposite side for one acute angle becomes the adjacent side for the other.

Applied problem

A ramp rises 4 feet over a horizontal distance of 12 feet.

If θ is the angle with the ground:

[ \tan\theta=\frac{4}{12}=\frac13 ]

If the question asks for tan θ, stop there.

Do not calculate θ unless needed. If the setup instead leaves you solving an equation for a side length, revisit our linear equations guide.

Calculator / Desmos

For exact special-angle questions, recognize structure.

For a non-special angle, calculator trig functions can help.

Make sure calculator angle mode matches the question:

• degrees vs radians.

Mini-practice

1

A right triangle has opposite side 9 and hypotenuse 15 relative to angle θ. What is (\sin\theta)?

2

A 45-45-90 triangle has legs of length 7. Hypotenuse?

3

A 30-60-90 triangle has hypotenuse 18. Short leg?

4

If (\sin 28^\circ = k), what is (\cos 62^\circ)?

5

A ladder reaches 12 feet up a wall and its base is 5 feet from the wall. If θ is the angle between the ladder and the ground, what is (\tan\theta)?

Answers

1. 3/5
2. (7\sqrt2)
3. 9
4. k
5. 12/5

Common traps

• using the wrong “opposite” side;
• confusing tangent with sine;
• forgetting special-triangle ratios;
• degree/radian calculator mismatch;
• solving for the angle when only a ratio is asked.

For more Geometry and Trigonometry skill guides, see our SAT Math guides.

Official references

• College Board Geometry and Trigonometry: https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/geometry-trigonometry

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