Trigonometric Ratios
Start gently
Trigonometric ratios are ratios between the sides of a right triangle. For the same angle the ratio never changes, however large the triangle — which is how you measure the height of a distant mountain.
Key points
Defining sin, cos, and tan
Take a right triangle and pick one of its acute angles, θ. Then sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. "Opposite" is the side across from θ, "adjacent" is the other short side.
The Values Worth Memorizing
Three angles come up constantly, so learn them cold. sin30°=1/2, cos30°=√3/2, tan30°=1/√3. sin45°=1/√2, cos45°=1/√2, tan45°=1. sin60°=√3/2, cos60°=1/2, tan60°=√3.
The Pythagorean Identity
sin²θ + cos²θ = 1 is the identity you will use most. It is the Pythagorean theorem in disguise, and it holds for every angle. Pair it with tan θ = sin θ ÷ cos θ and you can get any one ratio from another.
The Law of Sines
In any triangle ABC, each side divided by the sine of the angle across from it gives the same number — and that number is twice the radius R of the circle drawn through all three corners. Use it when you know two angles and a side.
The Law of Cosines
This one handles the cases the law of sines can't: find an angle when you know all three sides, or find the third side when you know two sides and the angle between them. Surveying and map-making run on it.
See it drawn
Turn the angle and the height rises and falls. This wave is what sound, radio waves and alternating current actually are.
Jobs that use this
Surveying Engineer$75k
Technical 3D Animator$90k
Telecommunications Engineer$100k
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