Right-Triangle Trigonometry UPCAT Reviewer: SOH-CAH-TOA and Special Angles

TEACHER ABI UPCAT MATHEMATICS

Right-Triangle Trigonometry and Special Angles

Choose the correct side ratio from a diagram, use special triangles efficiently, and model real height and distance situations without guessing the formula.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Right-Triangle Trigonometry and Special Angles

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Name the sides relative to the reference angle

The hypotenuse is always opposite the right angle. The names opposite and adjacent depend on the acute reference angle.

sinθ = oppositehypotenusecosθ = adjacenthypotenusetanθ = oppositeadjacent

SOH

sin = oppositehypotenuse

CAH

cos = adjacenthypotenuse

TOA

tan = oppositeadjacent

45°–45°–90°

Sides x:x:x√2

30°–60°–90°

Sides x:x√3:2x, opposite 30°, 60°, and 90°.

DO IT FAST

Circle what you know and what you need

1. Mark the reference angle.

2. Label O, A, and H relative to that angle.

3. Choose the ratio containing the known and unknown sides.

4. Check for 30°–60°–90° or 45°–45°–90° before reaching for a calculator.

Why it works

This makes the formula follow from the diagram and prevents using sine, cosine, or tangent based only on which one seems familiar.

WORKED EXAMPLES

Five forms you should recognize

1. Reference sides

Relative to θ, tangent uses the two legs: opposite÷adjacent.

2. 30°–60°–90°

Short leg x, long leg x√3, hypotenuse 2x.

3. 45°–45°–90°

Equal legs x give hypotenuse x√2.

4. Angle of elevation

tan35° = height40

COMMON TRAPS

Check before you commit

• Naming opposite and adjacent before choosing the reference angle
• Using diameter-style doubling in a triangle ratio
• Forgetting that the hypotenuse is opposite 90°
• Using sine when the known sides are opposite and adjacent
• Treating angle of depression as unrelated to angle of elevation
• Rounding too early in a multi-step problem

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

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Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

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QUICK ANSWERS

Right-Triangle Trigonometry and Special Angles FAQ

Is SOH-CAH-TOA used only for right triangles?

Yes, these basic side ratios in this form apply to right-triangle problems.

When should I use special-triangle ratios?

Use them when the acute angles are 30°, 45°, or 60° and exact values are desired.

Does the adjacent side include the hypotenuse?

No. “Adjacent” in SOH-CAH-TOA means the leg next to the reference angle, not the hypotenuse.

RELATED COMPETENCIES

Continue your mathematics review.

Pythagorean TheoremSimilar TrianglesCoordinate GeometryRadicals

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