Sine, cosine, and tangent

Why this matters

Pythagoras finds a side when you know two sides. Trigonometry goes further: it connects angles to sides. That is how surveyors measure a building's height from the ground, how engineers resolve a force into horizontal and vertical parts, how a roof pitch becomes a rafter length, and how games rotate a sprite to face a target.

The core idea

In a right triangle, pick one of the two acute angles and call it θ (theta). Relative to that angle, name the sides:

• Hypotenuse (H): the longest side, opposite the right angle.
• Opposite (O): the side across from θ.
• Adjacent (A): the side next to θ that is not the hypotenuse.

The labels depend on which angle you choose: the opposite side of one acute angle is the adjacent side of the other.

The three ratios are remembered as SOH-CAH-TOA:

For a given angle these ratios are fixed, however big the triangle is. sin 30° is always 0.5.

Finding a side

Step 1: label the sides relative to the known angle. Step 2: choose the ratio that uses the side you know and the side you want. Step 3: rearrange and compute.

Worked example 1: a ramp

A 10 m ramp rises at 30°. How high does it go?

• Known: hypotenuse = 10. Want: opposite (the height).
• O and H: use sine. sin 30° = O / 10.
• O = 10 × sin 30° = 10 × 0.5 = 5 m.

Worked example 2: a building's height

You stand 40 m from a building and the angle up to its roof is 52°.

• Known: adjacent = 40. Want: opposite.
• O and A: use tangent. O = 40 × tan 52° ≈ 40 × 1.280 = 51.2 m (plus your eye height, if you measured from your eyes).

Finding an angle

To go from a ratio back to an angle, use the inverse functions: sin⁻¹ (arcsin), cos⁻¹ (arccos) and tan⁻¹ (arctan).

Worked example 3: roof pitch

A roof rises 4 m over a horizontal run of 6 m. What angle does it make?

• O = 4, A = 6, so tan θ = 4 / 6 ≈ 0.667.
• θ = tan⁻¹(0.667) ≈ 33.7°.

Degrees and radians

Angles can be measured in degrees (a full turn is 360°) or radians (a full turn is 2π). Most programming languages, including Python and JavaScript, use radians in their trig functions. Convert first:

import math

print(math.sin(math.radians(30)))      # 0.49999999999999994, i.e. 0.5
print(40 * math.tan(math.radians(52)))  # about 51.2
print(math.degrees(math.atan(4 / 6)))  # about 33.69

The tiny error in the first line is normal floating-point rounding. Compare with a tolerance, not ==.

math.atan2(y, x) is a safer angle function in code: it handles all four directions and x = 0 without dividing by zero. Games use it to point an object towards a target.

Resolving a vector

A force of 100 N pulls at 30° above horizontal. Its parts are:

• horizontal = 100 × cos 30° ≈ 86.6 N
• vertical = 100 × sin 30° = 50 N

Splitting a quantity into x and y parts this way is the foundation of physics and engineering mechanics.

Common mistakes

• Calculator or code in the wrong angle mode. sin(30) in radians is about −0.988, not 0.5.
• Mislabelling opposite and adjacent. Always label from the angle you are using.
• Using trig ratios on a non-right triangle. SOH-CAH-TOA needs a 90° angle.
• Dividing when you should multiply. If the unknown is on top of the ratio, multiply; if it is on the bottom, divide (A = O / tan θ).

Quick recap

• Label O, A and H relative to the chosen angle.
• SOH-CAH-TOA: sin = O/H, cos = A/H, tan = O/A.
• Inverse functions turn a ratio back into an angle.
• Code uses radians; convert with math.radians and math.degrees.

Try it

In the Do tab, finish a function that finds the opposite side from an angle and the hypotenuse. An angle-finding function is already written as a model.