🔺 Lesson 1: Right Triangle Trigonometry

SOH-CAH-TOA — the foundation of vector components and 2D physics

🎯 Learning Objectives

Identify opposite, adjacent, hypotenuse relative to an angle
Apply SOH-CAH-TOA to find missing sides and angles
Calculate vector components using trigonometry
Solve incline plane and projectile launch problems

🔑 Key Concepts

SOH-CAH-TOA

Name	Ratio	Mnemonic
Sine	sin θ = opposite / hypotenuse	SOH
Cosine	cos θ = adjacent / hypotenuse	CAH
Tangent	tan θ = opposite / adjacent	TOA

Special Right Triangles — Memorize These Ratios!

Triangle	Angles	Side Ratios	sin/cos/tan Values
45°-45°-90°	45°, 45°, 90°	1 : 1 : √2	sin45°=cos45°=√2/2, tan45°=1
30°-60°-90°	30°, 60°, 90°	1 : √3 : 2	sin30°=½, cos30°=√3/2, tan30°=1/√3 sin60°=√3/2, cos60°=½, tan60°=√3

Inverse Trig Functions

When you know sides and need the angle:

θ = sin⁻¹(opposite/hypotenuse)
θ = cos⁻¹(adjacent/hypotenuse)
θ = tan⁻¹(opposite/adjacent)

Calculator Alert: Make sure your calculator is in DEGREE MODE (not radians) for these lessons. Physics problems typically use degrees for angles, radians for angular velocity.

📐 Worked Examples

Example 1: Finding Components — Force on an Incline

A 10 kg block sits on a 30° incline. Find the components of weight parallel and perpendicular to the plane.

Weight W = mg = 100 N. Angle with perpendicular = 30°.

W_parallel = W sin30° = 100 × 0.5 = 50 N (down the incline)
W_perp = W cos30° = 100 × (√3/2) = 86.6 N (into the plane)

Example 2: Finding Angle — Projectile Launch

A projectile has initial velocity components v_x = 20 m/s, v_y = 15 m/s. Find launch angle and speed.

tan θ = v_y / v_x = 15/20 = 0.75
θ = tan⁻¹(0.75) = 36.9°

v₀ = √(v_x² + v_y²) = √(400 + 225) = √625 = 25 m/s

Example 3: Incline with Friction

5 kg block on 25° incline, μ = 0.3. Find acceleration.

W = 50 N
W_∥ = 50 sin25° = 21.1 N
W_⊥ = 50 cos25° = 45.3 N = N (normal force)
f = μN = 0.3 × 45.3 = 13.6 N
F_net = W_∥ - f = 21.1 - 13.6 = 7.5 N
a = F_net/m = 7.5/5 = 1.5 m/s²

🏋️ Practice Problems

Right triangle: hypotenuse = 10, angle = 30°. Find other two sides.
Right triangle: adjacent = 6, opposite = 8. Find hypotenuse and both acute angles.
Force F = 50 N at 37° above horizontal. Find F_x and F_y. (sin37°≈0.6, cos37°≈0.8)
Projectile launched at 30 m/s, 45°. Find v_x, v_y.
Ramp angle 20°, box mass 10 kg. Find parallel/perpendicular weight components.
Two forces: F₁ = 30 N at 0°, F₂ = 40 N at 90°. Find resultant magnitude and direction.
Ladder 5m long leans against wall at 60°. How high up the wall? How far from wall at base?
sin θ = 0.6, θ in first quadrant. Find cos θ, tan θ.

✅ Answers

Click to reveal answers

1) opp=5, adj=5√3≈8.66 2) hyp=10, angles≈36.9°, 53.1° 3) F_x=40 N, F_y=30 N

4) v_x=v_y=21.2 m/s 5) W_∥=34.2 N, W_⊥=94.0 N 6) R=50 N, θ=53.1° 7) height=4.33m, base=2.5m

8) cos θ=0.8, tan θ=0.75 (3-4-5 triangle!)

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