Right Triangle Trigonometry

Basic Definitions (SOH-CAH-TOA)

For a right triangle with acute angle $\theta$:

$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \text{(SOH)}$

$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \text{(CAH)}$

$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \quad \text{(TOA)}$

And the reciprocals:

$\csc\theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{1}{\sin\theta}$

$\sec\theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{1}{\cos\theta}$

$\cot\theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{1}{\tan\theta}$

Pythagorean Theorem

For a right triangle with legs $a$ and $b$, and hypotenuse $c$:

$a^2 + b^2 = c^2$

45-45-90 Triangle (Isosceles Right Triangle)

Angles: $45°$, $45°$, $90°$

Side	Ratio
Leg	$1$
Leg	$1$
Hypotenuse	$\sqrt{2}$

If leg $= a$:

• Leg $= a$
• Leg $= a$
• Hypotenuse $= a\sqrt{2}$

If hypotenuse $= c$:

• Each leg $= \frac{c}{\sqrt{2}} = \frac{c\sqrt{2}}{2}$

Trig Values for 45°

Function	Value
$\sin 45°$	$\frac{\sqrt{2}}{2} \approx 0.707$
$\cos 45°$	$\frac{\sqrt{2}}{2} \approx 0.707$
$\tan 45°$	$1$
$\csc 45°$	$\sqrt{2}$
$\sec 45°$	$\sqrt{2}$
$\cot 45°$	$1$

30-60-90 Triangle

Angles: $30°$, $60°$, $90°$

Side	Ratio	Position
Short leg	$1$	Opposite $30°$
Long leg	$\sqrt{3}$	Opposite $60°$
Hypotenuse	$2$	Opposite $90°$

If short leg $= a$:

• Short leg (opp $30°$) $= a$
• Long leg (opp $60°$) $= a\sqrt{3}$
• Hypotenuse $= 2a$

Trig Values for 30°

Function	Value
$\sin 30°$	$\frac{1}{2} = 0.5$
$\cos 30°$	$\frac{\sqrt{3}}{2} \approx 0.866$
$\tan 30°$	$\frac{\sqrt{3}}{3} \approx 0.577$
$\csc 30°$	$2$
$\sec 30°$	$\frac{2\sqrt{3}}{3}$
$\cot 30°$	$\sqrt{3}$

Trig Values for 60°

Function	Value
$\sin 60°$	$\frac{\sqrt{3}}{2} \approx 0.866$
$\cos 60°$	$\frac{1}{2} = 0.5$
$\tan 60°$	$\sqrt{3} \approx 1.732$
$\csc 60°$	$\frac{2\sqrt{3}}{3}$
$\sec 60°$	$2$
$\cot 60°$	$\frac{\sqrt{3}}{3}$

Solving Right Triangles

Given information, find all sides and angles.

Given Two Sides

1. Use Pythagorean theorem for third side
2. Use trig ratios for angles

Given One Side and One Acute Angle

1. Use trig ratios to find other sides
2. Subtract from $90°$ for other acute angle

Applications

Angle of Elevation

Angle measured upward from horizontal.

Angle of Depression

Angle measured downward from horizontal.

Key insight: Angle of elevation from A to B equals angle of depression from B to A (alternate interior angles).

Quick Reference: Special Angle Values

Angle	$\sin$	$\cos$	$\tan$
$0°$	$0$	$1$	$0$
$30°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$
$45°$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$
$60°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$
$90°$	$1$	$0$	undefined