Introduction to Trigonometry

What is Trigonometry?

Trigonometry: Study of relationships between angles and sides in triangles

Focus: Right triangles

Three main ratios:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)

Parts of a Right Triangle

Relative to an angle θ:

Hypotenuse: Longest side (opposite right angle)

Opposite side: Side across from angle θ

Adjacent side: Side next to angle θ (not hypotenuse)

Example: Identify Sides

For angle A:

• Side BC = opposite
• Side AB = adjacent
• Side AC = hypotenuse

For angle C:

• Side AB = opposite
• Side BC = adjacent
• Side AC = hypotenuse

Sides change based on which angle you're using!

The Three Trigonometric Ratios

SOH-CAH-TOA (memory trick!)

Sine: sin(θ) = Opposite / Hypotenuse

Cosine: cos(θ) = Adjacent / Hypotenuse

Tangent: tan(θ) = Opposite / Adjacent

Remember:

• Sin = Opp / Hyp (SOH)
• Cos = Adj / Hyp (CAH)
• Tan = Opp / Adj (TOA)

Finding Trig Ratios

Example 1: 3-4-5 Triangle

Sides: 3 (opp), 4 (adj), 5 (hyp)

Find sin(θ):

sin(θ) = opp/hyp = 3/5

Find cos(θ):

cos(θ) = adj/hyp = 4/5

Find tan(θ):

tan(θ) = opp/adj = 3/4

Example 2: 5-12-13 Triangle

Angle A: opp = 5, adj = 12, hyp = 13

Ratios:

sin(A) = 5/13
cos(A) = 12/13
tan(A) = 5/12

Example 3: Using Pythagorean Theorem

Given: opp = 7, hyp = 25

Find adj:

7² + adj² = 25²
49 + adj² = 625
adj² = 576
adj = 24

Now find ratios:

sin(θ) = 7/25
cos(θ) = 24/25
tan(θ) = 7/24

Using Calculator for Angles

Find ratio from angle:

Example: sin(30°)

• Calculator: sin(30) = 0.5

Example: cos(45°)

• Calculator: cos(45) ≈ 0.707

Example: tan(60°)

• Calculator: tan(60) ≈ 1.732

Make sure calculator is in DEGREE mode!

Finding Angle from Ratio

Inverse trig functions:

• sin⁻¹ (or arcsin)
• cos⁻¹ (or arccos)
• tan⁻¹ (or arctan)

Example: Find Angle

If sin(θ) = 0.5, find θ

Use: θ = sin⁻¹(0.5) = 30°

If tan(θ) = 1, find θ

Use: θ = tan⁻¹(1) = 45°

Finding Missing Sides

Given angle and one side, find another side

Example 1: Find Opposite

Angle = 35°, Hypotenuse = 20 Find opposite side

Use sin:

sin(35°) = opp/20
opp = 20 × sin(35°)
opp = 20 × 0.574
opp ≈ 11.5

Answer: ≈ 11.5

Example 2: Find Adjacent

Angle = 50°, Hypotenuse = 15 Find adjacent side

Use cos:

cos(50°) = adj/15
adj = 15 × cos(50°)
adj = 15 × 0.643
adj ≈ 9.6

Answer: ≈ 9.6

Example 3: Find Hypotenuse

Angle = 25°, Opposite = 8 Find hypotenuse

Use sin:

sin(25°) = 8/hyp
hyp = 8/sin(25°)
hyp = 8/0.423
hyp ≈ 18.9

Answer: ≈ 18.9

Finding Missing Angles

Given two sides, find angle

Example 1: Find Angle

Opposite = 6, Hypotenuse = 10 Find angle

Use sin:

sin(θ) = 6/10 = 0.6
θ = sin⁻¹(0.6)
θ ≈ 36.9°

Answer: ≈ 36.9°

Example 2: Using Tangent

Opposite = 5, Adjacent = 12 Find angle

Use tan:

tan(θ) = 5/12
θ = tan⁻¹(5/12)
θ ≈ 22.6°

Answer: ≈ 22.6°

Special Angles

Memorize these (from special right triangles):

30° angle:

• sin(30°) = 1/2
• cos(30°) = √3/2
• tan(30°) = 1/√3 = √3/3

45° angle:

• sin(45°) = √2/2
• cos(45°) = √2/2
• tan(45°) = 1

60° angle:

• sin(60°) = √3/2
• cos(60°) = 1/2
• tan(60°) = √3

Complementary Angles

In right triangle, two acute angles are complementary (sum to 90°)

Property: sin(θ) = cos(90° - θ)

Example:

• sin(30°) = cos(60°) = 1/2
• sin(60°) = cos(30°) = √3/2

Real-World Applications

Architecture: Building angles and heights

• Find height of building from distance and angle

Navigation: Bearing and distance

• Determine position using angles

Engineering: Slope of ramps

• Calculate angle from rise and run

Astronomy: Calculating distances

• Using angles to stars and planets

Sports: Projectile angles

• Optimal angle for throwing/kicking

Example: Height of Tree

Standing 50 ft from tree, angle to top is 35°

Find height:

tan(35°) = height/50
height = 50 × tan(35°)
height = 50 × 0.700
height ≈ 35 ft

Answer: Tree is about 35 ft tall

Angle of Elevation and Depression

Angle of elevation: Looking up from horizontal

Angle of depression: Looking down from horizontal

Both measured from horizontal line

Example: Angle of Depression

From 100 ft tower, see object with 20° angle of depression

Find distance from base:

tan(20°) = 100/distance
distance = 100/tan(20°)
distance ≈ 275 ft

Practice