Same angle, same shape
Draw a right triangle with a 30° angle. Draw another, ten times bigger, with the same 30° angle. The second is a perfect enlargement of the first: every side ten times longer. So the ratios between the sides are identical. They depend only on the angle, never on the size.
That's the whole idea of trigonometryThe mathematics of how the angles of a triangle relate to the lengths of its sides.: those ratios get names, and a calculator knows them for every angle. Name the sides relative to the angle (“theta”) you're looking at:
- hypotenuse: the long side, opposite the right angle.
- opposite: the side across from .
- adjacent: the side next to that isn't the hypotenuse.
The three ratios
Say it out loud: “sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.”
The angle· measured in degrees (°)
The corner you're measuring from (not the right angle).
Sine
What fraction of the hypotenuse the opposite side is.
Cosine
What fraction of the hypotenuse the adjacent side is.
Tangent
The slope of the hypotenuse: opposite per adjacent.
Many people remember them as SOH-CAH-TOA: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent. Change the angle and the size below, and watch which numbers move:
Triangle view: a right triangle with angle θ at the bottom left and a hypotenuse of length r, both set by sliders, with the opposite and adjacent sides labelled. The ratios opposite over hypotenuse (sine), adjacent over hypotenuse (cosine) and opposite over adjacent (tangent) depend only on the angle. Circle view: a point on a circle of radius 1 at angle θ, from 0 to 360 degrees; its across position is cos θ and its height is sin θ.
One angle, one shape
InteractiveView
The math, with your numbers
Change r: the sides change, the three ratios don't.
Try this
- →At what angle are the opposite and adjacent sides equal? What's tan θ there?
- →Find the angle where sin θ = 0.5.
- →In circle view, go past 90°. Which of sin or cos turns negative first?
Splitting an arrow into across and up
This is what physics uses trigonometry for, constantly. An arrow of length at angle above the ground is the hypotenuse of a right triangle. Rearranging the definitions gives its two components:
Worked example: a ball kicked at 20 m/s, 30° above the ground
Step 1 of 4- 1The launch velocity is the hypotenuse; the across and up parts are the adjacent and opposite sides.
Going backwards: finding the angle
If you know the sides and want the angle, use the inversesin⁻¹, cos⁻¹ and tan⁻¹ (also written arcsin, arccos, arctan): they take a ratio and give back the angle. functions, written , and . They answer “which angle has this ratio?”.
Make a prediction first
A ramp rises 1 m over a horizontal distance of 4 m. How steep is it, in degrees?
Think about it, then click to reveal the answer.
Make a prediction first
A ramp rises 1 m over a horizontal distance of 4 m. How steep is it, in degrees?
Think about it, then click to reveal the answer.
You know the opposite (1 m) and the adjacent (4 m), so use the tangent: .
Wrap-up
Key ideas to remember
- In a right triangle, the ratios of the sides depend only on the angle.
- SOH-CAH-TOA: , , .
- Components of an arrow: , .
- Inverse functions (, , ) turn a ratio back into an angle.
Check your understanding
Sources & further reading
Everything in this lesson agrees with these references. They're all free to read.