Algebra 101 - Unit 15

Lesson 93: Trigonometric Functions (Sine, Cosine, Tangent)

Trigonometric functions relate angles to ratios of sides in a right triangle or to coordinates on the unit circle. The three primary functions are sine (sin), cosine (cos), and tangent (tan).

Definitions

For a right triangle with angle θ:

• Sine: sin θ = opposite / hypotenuse
• Cosine: cos θ = adjacent / hypotenuse
• Tangent: tan θ = opposite / adjacent = sin θ / cos θ

On the unit circle (radius = 1), a point (x, y) corresponding to angle θ satisfies:

• cos θ = x-coordinate
• sin θ = y-coordinate
• tan θ = y / x (undefined when x = 0)

Examples

Example 1 — Right Triangle: In a triangle with hypotenuse 10 and opposite side 6 for angle θ, find sin θ, cos θ, tan θ.

• sin θ = 6/10 = 0.6
• cos θ = √(10² − 6²)/10 = √64/10 = 8/10 = 0.8
• tan θ = 6/8 = 0.75

Example 2 — Unit Circle: θ = 150° → point on circle = (−√3/2, 1/2)

• cos 150° = −√3/2
• sin 150° = 1/2
• tan 150° = (1/2) / (−√3/2) = −1/√3 ≈ −0.577

Practice Problems

1. For a right triangle with opposite = 5 and adjacent = 12, find sin θ, cos θ, tan θ.
2. Find sin, cos, and tan for θ = 45° using the unit circle.
3. Determine the coordinates for θ = 210° on the unit circle and find sin θ, cos θ, tan θ.
4. Convert tan θ = √3/3 to the corresponding acute angle in degrees.
5. A triangle has hypotenuse 13 and opposite side 5. Find cos θ and tan θ.