Question

How do you find the sine, cosine, and tangent values on the unit circle? Provide an example.

Answer 1

In the unit circle sine and cosine are given by the y-coordinate and x-coordinate, respectively. Tangent is the ratio y-coordinate / x-coordinate.

That is so in virtue of the unit circle has radius 1. That implies that the hypotenuse is 1, so sine is opposite-leg / hypotenuse = y-ccordinate / 1; and cosine is adjacent-leg / hypotenuse = x-coordinate / 1.

Some examples are:

1) Angle 0 ⇒ x-coordinate = 1, y-coordinate ⇒ 

cosine 0 = 1; 
sine 0 = 0; 
tangent = 0 / 1 = 0


2) Angle 45° ⇒ x-coordinate = (√2 ) / 2; y-coordinate = (√2) / 2 ⇒

cosine 45° = (√2 / 2)
sine 45° = (√2 ) / 2

tan 45° = 1

3) Angle 90° ⇒ x-coordinate = 0; y-coordinate = 1 ⇒

cos 90° = 0
sin 90° = 1
tan 90° = 1 / 0 = ∞ (undefined)

Answer 2

To find the sine, cosine, and tangent values on the unit circle:

• cos a = coordinate value x
• sin a = coordinate value of y
• tan a = coordinate value y : coordinate value x

Further explanation

In trigonometry, there are 3 angular functions, namely sin, cos and tan, based on a right triangle

There are 3 sides used in this triangle, namely:

• 1. hypotenuse: the longest side
• 2. opposite to the angle θ
• 3. adjacent to the angle θ

General formulas for sin, cos, and tan:

$\displaystyle sin\:a=\frac{opposite}{hypotenuse}$

$\displaystyle cos\:a=\frac{adjacent}{hypotenuse}$

$\displaystyle tan\:a=\frac{opposite}{adjacent}$

There are 3 other functions:

$\displaystyle sec\:a=\frac{1}{cos\:a}$

$\displaystyle csc\:a=\frac{1}{sin\:a}$

$\displaystyle cot\:a=\frac{1}{tan\:a}$

The unit circle is a circle with a radius = 1

Because of the value of r = 1, the value :

• cos a = coordinate value x
• sin a = coordinate value of y
• tan a = coordinate value y : coordinate value x

So the coordinate point in the circle is a point (cos a, sin a)

A Pythagorean theorem which states that:

the hypotenuse or the longest side in a right triangle equal to the sum of the squares of the other sides.

Then in the unit circle, the phytagorean equation becomes :

$\displaystyle x^2+y^2=1^2$

$\displaystyle x^2+y^2=1$

Example

• the angle θ = 0 °, x = 1, y = 0

$\displaystyle sin\:a=\frac{opposite}{hypotenuse}=\frac{0}{1}=0$

$\displaystyle cos\:a=\frac{adjacent}{hypotenuse}=\frac{1}{1}=1$

$\displaystyle tan\:a=\frac{opposite}{adjacent}=\frac{0}{1}=0$

• the angle θ = 90, x = 0, y = 1

$\displaystyle sin\:a=\frac{opposite}{hypotenuse}=\frac{1}{1}=1$

$\displaystyle cos\:a=\frac{adjacent}{hypotenuse}=\frac{0}{1}=0$

$\displaystyle tan\:a=\frac{opposite}{adjacent}=\frac{1}{0}=undfned$

Learn more

Pythagoras theorem

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 trigonometric ratio to find x

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trigonometric identities

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Keywords: unit circle, coordinate point, sin, cos, tan