Trigonometric Ratios

SOH CAH TOA

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

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \quad \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \quad \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

Memory trick: Some Old Horse Caught Another Horse Taking Oats Away

Special Right Triangles

45-45-90 Triangle

Legs are equal; hypotenuse is $\sqrt{2}$ times a leg.

$1 : 1 : \sqrt{2}$

If leg $= a$: hypotenuse $= a\sqrt{2}$

30-60-90 Triangle

$1 : \sqrt{3} : 2$

• Short leg (opposite 30°) $= a$
• Long leg (opposite 60°) $= a\sqrt{3}$
• Hypotenuse (opposite 90°) $= 2a$

Finding Missing Sides

Given: angle $\theta = 35°$ and hypotenuse $= 10$

$\text{Opposite} = 10 \sin 35° \approx 5.74$ $\text{Adjacent} = 10 \cos 35° \approx 8.19$

Finding Missing Angles

Use inverse trig functions:

$\theta = \sin^{-1}\left(\frac{\text{opp}}{\text{hyp}}\right) \quad \theta = \cos^{-1}\left(\frac{\text{adj}}{\text{hyp}}\right) \quad \theta = \tan^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)$

Example: Opposite $= 5$, Adjacent $= 12$ $\theta = \tan^{-1}\left(\frac{5}{12}\right) \approx 22.6°$

Angles of Elevation and Depression

• Elevation: Looking UP from horizontal
• Depression: Looking DOWN from horizontal

Both form right triangles with the horizontal ground.

Example: A 50 ft building, angle of elevation $= 40°$. Distance from base: $\tan 40° = \frac{50}{d} \implies d = \frac{50}{\tan 40°} \approx 59.6 \text{ ft}$

Complementary Angle Relationship

$\sin \theta = \cos(90° - \theta) \quad \text{and} \quad \cos \theta = \sin(90° - \theta)$

Tip: Always label which side is opposite, adjacent, and hypotenuse relative to the angle you're working with!