Sin, cos and tan for sides
Name the sides from the angle
Mark the angle you are using, θ.
Then:
Not drawn accurately
The hypotenuse is opposite the right angle. The opposite is across the triangle from θ. The adjacent is next to θ and is not the hypotenuse.
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
The first letters spell SOH CAH TOA.
Find a side in three steps
- Label the side you know and the side you want. 2. Pick the ratio that uses those two. 3. Write the equation and rearrange.
- angle 35°, hypotenuse 12 cm, want the opposite x
- opposite and hypotenuse: use sin
- sin 35° = x/12
- x = 12 × sin 35° = 6.88... = 6.9 cm (1 d.p.)
If x is on the bottom of the fraction you divide instead: tan 40° = 9/x gives x = 9 ÷ tan 40° = 10.7 cm.
Degrees, not radians
Calculators can work in degrees or radians.
In radians sin 35 = −0.428, which makes no sense for a length. Check your calculator shows D or DEG before any trigonometry.
Sin and cos on the unit circle
The circle has radius 1, so its radius is a hypotenuse of length 1. Type 40 as the angle: the vertical side of the triangle is sin 40° = 0.643 and the horizontal side is cos 40° = 0.766. A ladder 5 m long at 40° to the ground is five times as big, so its top is 5 × 0.643 = 3.2 m up. Now try 70°: which side grows, and which shrinks?
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