Introduction to Trigonometry
Grade 9 · Math · Study Guide
Learn the three basic trig ratios (sine, cosine, tangent), apply them to right triangles, and use SOH-CAH-TOA to find missing sides and angles.
Trigonometry studies the relationships between the angles and sides of triangles. In a right triangle, we label the sides relative to a reference angle θ (theta): the side opposite the angle, the side adjacent to the angle, and the hypotenuse (always opposite the right angle). The three main trigonometric ratios are:
• Sine: sin(θ) = opposite / hypotenuse • Cosine: cos(θ) = adjacent / hypotenuse • Tangent: tan(θ) = opposite / adjacent
The memory aid SOH-CAH-TOA helps you remember these: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. For example, in a right triangle where θ = 30°, the sides are in the ratio 1:√3:2 (opposite:adjacent:hypotenuse).
To find a missing side, set up the appropriate ratio and solve. To find a missing angle, use the inverse trig functions: sin⁻¹, cos⁻¹, or tan⁻¹ on a calculator. Special angle values (30°, 45°, 60°) appear frequently and are worth memorizing.
| Angle (θ) | 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
- Sine (sin)
- In a right triangle: opposite side / hypotenuse
- Cosine (cos)
- In a right triangle: adjacent side / hypotenuse
- Tangent (tan)
- In a right triangle: opposite side / adjacent side
- SOH-CAH-TOA
- Memory aid for trig ratios: Sine=Opp/Hyp, Cosine=Adj/Hyp, Tangent=Opp/Adj
- Theta (θ)
- A variable commonly used to represent an unknown angle
- Inverse trig functions
- sin⁻¹, cos⁻¹, tan⁻¹ — used to find an angle given a ratio
Flashcards
- SOH-CAH-TOA
- sin=opp/hyp, cos=adj/hyp, tan=opp/adj
- Sine
- sin(θ) = opposite / hypotenuse
- Cosine
- cos(θ) = adjacent / hypotenuse
- Tangent
- tan(θ) = opposite / adjacent
- Hypotenuse
- Longest side of a right triangle; opposite the right angle
- Inverse trig (sin⁻¹, cos⁻¹, tan⁻¹)
- Calculator functions that return the angle when you know the ratio
Practice questions
In a right triangle, sin(θ) equals:
- adjacent / hypotenuse
- opposite / hypotenuse
- opposite / adjacent
- hypotenuse / opposite
Show answer
Answer: opposite / hypotenuse
By definition, sin(θ) = opposite ÷ hypotenuse (the 'S' in SOH).
tan(45°) = 1
Show answer
Answer: True
tan(45°) = sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1.
The memory aid for trig ratios is ___-CAH-TOA.
Show answer
Answer: SOH
SOH stands for Sine = Opposite / Hypotenuse.
A right triangle has an angle of 30° and a hypotenuse of 10. The side opposite 30° is:
- 5
- 8.66
- 10
- 2.5
Show answer
Answer: 5
sin(30°) = 0.5; opposite = sin(30°) × hypotenuse = 0.5 × 10 = 5.
Which trig ratio uses the opposite and adjacent sides?
- sine
- cosine
- tangent
- hypotenuse
Show answer
Answer: tangent
Tangent = opposite / adjacent (the 'TOA' in SOH-CAH-TOA).
Fun facts
- Trigonometry was invented by ancient Greek astronomers to measure the distances to the Moon and Sun without leaving Earth.
- Every GPS satellite uses trigonometry to calculate your exact position by triangulating signals from multiple satellites.
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