Trigonometric Ratios
Grade 10 · Math · Study Guide
Using sine, cosine, and tangent to solve right triangle problems.
Trigonometry literally means 'triangle measurement,' and the three primary ratios — sine, cosine, and tangent — allow you to find unknown side lengths and angles in right triangles. The acronym SOH-CAH-TOA summarizes them: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. These ratios are defined relative to one of the acute angles in the right triangle.
The hypotenuse is always the longest side, directly across from the right angle. The 'opposite' side is across from the angle you're working with, and the 'adjacent' side is next to that angle (but not the hypotenuse). Labeling sides correctly before writing any ratio is critical — a common source of errors.
To find a missing side, identify which ratio connects your known and unknown values, write the equation, then solve algebraically. For example, if you know an angle of 35° and the hypotenuse is 10, to find the opposite side: sin(35°) = x/10, so x = 10·sin(35°) ≈ 5.74.
To find a missing angle, use inverse trig functions: sin⁻¹, cos⁻¹, or tan⁻¹ (also written arcsin, arccos, arctan). If tan(θ) = 3/4, then θ = tan⁻¹(3/4) ≈ 36.87°. The Pythagorean theorem (a² + b² = c²) works alongside trig ratios — if you know two sides and need the third, the Pythagorean theorem is often faster.
| Angle | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 ≈ 0.577 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 ≈ 1.732 |
| 90° | 1 | 0 | undefined |
Key terms
- SOH-CAH-TOA
- Memory device: Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj.
- Hypotenuse
- The longest side of a right triangle; always opposite the right angle.
- Opposite side
- The side directly across from the reference angle (not the hypotenuse).
- Adjacent side
- The side next to (touching) the reference angle, not the hypotenuse.
- Inverse Trig Function
- sin⁻¹, cos⁻¹, tan⁻¹; used to find an unknown angle from a known ratio.
Flashcards
- SOH
- Sine = Opposite / Hypotenuse
- CAH
- Cosine = Adjacent / Hypotenuse
- TOA
- Tangent = Opposite / Adjacent
- Inverse trig functions
- sin⁻¹, cos⁻¹, tan⁻¹ — find the angle when you know the ratio
- sin 30°
- 1/2 = 0.5
- cos 60°
- 1/2 = 0.5
Practice questions
In a right triangle, sin(θ) is defined as:
- Adjacent/Hypotenuse
- Opposite/Adjacent
- Opposite/Hypotenuse
- Hypotenuse/Opposite
Show answer
Answer: Opposite/Hypotenuse
SOH: Sine = Opposite/Hypotenuse. This is the foundational definition.
In SOH-CAH-TOA, TOA stands for Tangent = Opposite / ___.
Show answer
Answer: Adjacent
TOA: Tangent = Opposite/Adjacent. Tangent compares the two legs (not the hypotenuse).
The hypotenuse is always the side opposite the right angle in a right triangle.
Show answer
Answer: True
The hypotenuse is always directly opposite the 90° angle and is always the longest side.
If cos(θ) = 0.6 and the hypotenuse is 15, what is the adjacent side?
- 6
- 9
- 12
- 25
Show answer
Answer: 9
cos(θ) = adj/hyp → 0.6 = adj/15 → adj = 0.6 × 15 = 9.
To find a missing angle in a right triangle when two sides are known, you use an inverse trig function.
Show answer
Answer: True
Inverse trig functions reverse the ratio: if you know sin, cos, or tan of the angle, the inverse function gives you the angle.
Fun facts
- Ancient Greek astronomers used early trigonometry to estimate the distance to the Moon with remarkable accuracy — over 2,000 years before GPS.
- Every video game that renders a 3D world uses trigonometric functions thousands of times per frame to calculate positions, lighting angles, and collision detection.
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