Right-angle triangles, sine and cosine

Pythagoras

Greek Mathematician, philosopher and scientist, 570 - 495 BCE. Best-known for his Pythagorean Theorem, which defines the lengths of sides of a right-angled triangle. Leader of a mystic cult proposi

A right-angled triangle has two 'legs', the short sides, and the longest side, called the hypotenuse. The sum of the squares of the legs equals the square of the hypotenuse.

In similar triangles, the side lengths may be different, but the angles remain the same. This means that the ratios of the side lengths will be the same. There are special functions which describe these unchanging ratios of side lengths: sine and cosine.

The lengths of the sides of a right-angled triangle and the internal angles are related by the functions sine, cosine, and tan

In a right-angled triangle, the lengths of the sides may be expressed as a trigonometric function of the angles.

sinθ = b/c
cosθ = a/c
tanθ = b/a

Learn to recite:

SOH CAH TOA

SOH = Sine is the Opposite over the Hypotenuse

CAH = Cosine is the Adjacent over the Hypotenuse

TOA = Tan is the Opposite over the Adjacent

This is a neat way to remember the sin, cos and tan functions:

SOH: sinθ = ${\opposite}/{\hypotenuse} = O/H$

CAH: cosθ = ${\adjacent}/{\hypotenuse} = A/H$

TOA: tanθ = ${\opposite}/{\adjacent} = O/A$

The Sine Function

The sine function is often written in the short form 'sin', but is pronounced like 'sign'.

sinθ = ${\opposite}/{\hypotenuse} = O/H$

Cosine Function

The cosine function also has a short form 'cos'.

cosθ = ${\adjacent}/{\hypotenuse} = A/H$

Tan Function

The 'tan' function is short for 'tangent', but it is not the same as the tangent of a circle.

tanθ = ${\opposite}/{\adjacent} = O/A$

Some commonly needed values of these functions are:

θ	sine	cosine	tan
0°	0	$$1$$	$$0$$
30°	$${1}/{2}$$	$${√3}/2$$	$${1}/{√3}$$
45°	$${1}/{√2}$$	$${1}/{√2}$$	$$1$$
60°	$${√3}/{2}$$	$${1}/{2}$$	$${√3}$$
90°	$${0}$$	$${1}$$	$${-}$$

sin(B̂) = cos(90 - B̂)

What are the sizes of the other angles in the triangle? One is a right angle, 90°. Since the total of all the angles in a triangle is 180°, B̂ and Â must sum to 90°, or Â = 90° - B̂.

What is sin(90°- B̂)? It is ${\opposite}/{\hypotenuse} = a/c$.

We saw that cosB̂ = $a/c$, so we have discovered that cosB = sin(90°- B̂).

Similarly, sinB = cos(90°- B̂).

arcsin and arccos

We know that if we have an angle and the lengths of two sides to a triangle, we can calculate the third length, using sin, cos, or tan, depending on which sides we know.

But, what if we know two sides, such as $b$ and $c$, and we want to know the angle B̂?

sinB̂ = $a/c$, so B̂= arcsin$(a/c)$.

arcsin is a function which finds the angle B̂ from the value of sinB̂.

Similarly, if you know cosB̂ or tanB̂, you can find B̂ from arccosB̂ or arctanB̂.

You will find these functions on your calculator as $sin^{-1}$, $cos^{-1}$, and $tan^{-1}$.

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