The Foundation of Right Triangle Trigonometry

What Are Acute Angles?

Definition: Angles measuring between 0° and 90° (exclusive)
In radians: between 0 and $$\frac{\pi}{2}$$ (exclusive)

Key Property: Acute angles are the angles in right triangles (besides the 90° angle).

Setting Up the Right Triangle

For an acute angle θ in right triangle ABC (with C = 90°):

        B
        /|
       / |
    c /  | a (opposite to θ)
     /   |
    /____|
   A   b   C
   (adjacent to θ)

Labels:

• a = side opposite angle θ (across from θ)

• b = side adjacent to angle θ (next to θ, not the hypotenuse)

• c = hypotenuse (longest side, opposite right angle)

The Six Trigonometric Ratios

Primary Three (SOH-CAH-TOA):

1. Sine:

$$\sin\theta$$=$$\frac{opposite}{hypotenuse}$$​

2. Cosine:

$$\cos\theta$$​=$$\frac{adjacent}{hypotenuse}$$

3. Tangent:

$$\tan\theta$$​=$$\frac{opposite}{adjacent}$$

Reciprocal Three:

4. Cosecant (reciprocal of sine):

$$\csc\theta$$=$$\frac{hypotenuse}{opposite}$$​

5. Secant (reciprocal of cosine):

$$\sec\theta$$​=$$\frac{hypotenuse}{adjacent}$$​

6. Cotangent (reciprocal of tangent):

$$\cot\theta$$​=$$\frac{adjacent}{opposite}$$​

The Complementary Angle Relationship

In a right triangle, the two acute angles are complementary (sum to 90° or π/2​).

If α and β are complementary (α + β = 90°):

$$\sin\alpha=\cos\beta , \tan\alpha=\cot\beta , \sec\alpha=\csc\peta$$

In function notation:

sin⁡(90°−θ)=cos⁡θ , cos⁡(90°−θ)=sin⁡θ , tan⁡(90°−θ)=cot⁡θ

Exact Values for Special Acute Angles

30°-60°-90° Triangle (π/6 – π/3 – π/2)

Side ratios: 1 : √3 : 2

        60°
        /|
       / |
      /  |
   2 /   | √3
    /    |
   /_____|
 30°   1

Values:

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

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

45°-45°-90° Triangle (π/4 – π/4 – π/2)

Isosceles right triangle, side ratios: 1 : 1 : √2

       45°
       /|
      / |
     /  |
 √2 /   | 1
   /    |
  /_____|
45°   1

Values:

• sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1

Table of Exact Values

Solving Right Triangles

Given some parts, find all remaining sides and angles.

Types of Problems:

1. Given: Two sides → find third side (Pythagorean theorem) and angles

2. Given: One side and one acute angle → find other sides and angle

Example 1: Given two sides

Triangle with sides 5 and 12, right angle between them

1. Hypotenuse: c=$$\sqrt{5^2+12^2}=\sqrt{169}=13$$

2. Find angle θ opposite side 5:
   sin⁡θ=$$\frac{5}{13}$$ ⇒ θ=arcsin⁡($$\frac{5}{13}$$)≈22.62°

3. Other angle: 90° – 22.62° = 67.38°

Example 2: Given side and angle

Hypotenuse = 10, angle = 35°

1. Find side opposite 35°: $a=10\sin 35°\approx 10\times 0.5736=5.736$

2. Find side adjacent to 35°: $b=10\cos 35°\approx 10\times 0.8192=8.192$

3. Other angle: 90° – 35° = 55°

Applications of Right Triangle Trig

A. Angle of Elevation/Depression

• Angle of elevation: Looking UP from horizontal

• Angle of depression: Looking DOWN from horizontal

• These angles are equal if lines are parallel (alternate interior angles)

Example: A 50ft tree casts a 30ft shadow. Find sun’s angle of elevation.

tan⁡θ=$$\frac{50}{30}=\frac{5}{3}$$⇒θ=arctan⁡($$\frac{5}{3}$$)≈59.04°

Bearing/Navigation Problems

Bearing: Angle measured clockwise from North

• N30°E = 30° east of north

• S45°W = 45° west of south