The unit circle

Define using the coordinate system a circle whose centre is at the origin and whose radius is 1.

From the course 'Analytical Geometry', we remember that the equation of such a circle is

Oriented angles

Select point A on the circumference of the unit circle.

The angle between the radius drawn at point A and the x-axis is 𝛼.

Form a right triangle such that the hypotenuse

is the radius of the circle.

The coordinates of point A are (0,8; 0,6). Since one vertex of the triangle is at the origin, the values of the coordinates of the circumferential point A are formed directly as the lengths of the catheters of the triangle. In this case, the lengths of the catheters are 0,8 and 0,6, and the length of the hypotenuse is 1.

Now we could find the angle 𝛼 using the trigonometric functions sine or cosine

That is, sin (36,9 °) = 0,6 and cos (36,9 °) = 0,8 We could indicate the coordinates of point A using the angle 𝛼 A (cos (36,9 °), sin (36,9 °)) .

Let (x, y) be the coordinates of the circumferential point A. Then the angle 𝛼 holds

Example 2

Find the coordinates of the circumferential points A, B and C.

The angle corresponding to point A is 30 °

The angle corresponding to point B is 130 °

The angle corresponding to point C is 240 °

Based on the previous example

A (cos (30 °), sin (30 °)), B (cos (130 °), sin (130 °)), C (cos (240 °), sin (240 °))

We calculate the coordinates to one decimal place.

A (0,9;0,5)

B (-0,6;0,8)

C (-0,5;-0,9)

Properties of sine and cosine

It has been defined above that sine and cosine are the coordinates of the circumference point on a unit circle. The values of sine and cosine thus vary between [-1,1]. Sine and cosine get all the values between [-1,1].

For all the values of the angle 𝛼

The sign of sine and cosine is determined by which quarter of the coordinate system the perimeter point is located. Sine is the y-coordinate and cosine is the x-coordinate.

Sine

Cosine

Example 3

At what angle sin (𝛼) = 0,5?

Since the y-coordinate is 0,5 at two circumferential points, there must be two angles with a sine value of 0,5. The circumferential points A and B with a y-coordinate of 0,5 are marked below.

An angle equal to the angle 𝛼 is formed between the negative x-axis and the line segment drawn at point B. We could find this using similarity. Equal angles are marked in the figure below.

So it has to be that

We solve the angle at which the value of the sine is 0,5

At an angle of 30°, the value of sine is 0,5. According to the above, also at an angle of 180° - 30° = 150°, the sine should have a value of 0,5.

Example 4

What angle is 𝛼

The cosine is the x-coordinate of the circumferential point and the two circumferential points have an x-coordinate value equal to the requested value.

This forms a corresponding triangle below the x-axis so angle 𝛼 is equal to the angle 𝛽. The angle rotating clockwise is denoted as negative, so 𝛽 = -𝛼.

Then between 0° and 360° the cosine would have the same values with

angles 𝛼 and 360° - 𝛼

Coterminal angles and supplementary angles

Coterminal angles

Supplementary angle

Periodicity

If a multiple of the full angle 360° is added to the angle, the circumference point remains unchanged. That is, sine and cosine always has the same value every 360°.

where n is an integer. This is called the periodicity of sine and cosine.

Pythagorean trigonometric identity

The radius drawn from the origin to the circumferential point of the unit circle forms a right triangle in which the radius is the hypotenuse. The catheters are x and y. In this case, Pythagorean theorem states

This is also the equation of an origin-centred circle with a radius of 1.

Since cos (𝛼) = x and sin (𝛼) = y, we get

This is Pythagorean trigonometric identity

Radians

A full angle is 360° and the circumference of the unit circle is 2𝝅. Let us introduce the unit of an angle, the radian, where 2𝝅 rad = 360 °.

All of the above properties of sine and cosine also apply when the unit of an angle is a radian.

Example 5

From [0,2𝝅], find the angles that have the same value as

The period of sine and cosine is 2𝝅, that is, every 2𝝅 sine and cosine has the same value. In addition to this, sine has the same value at supplement angles and cosine at coterminal angles.

a) sin (𝝅 / 4) = sin (𝝅-𝝅 / 4), the angle at which the sine has the same value as the angle 𝝅 / 4

b) cos (2𝝅 / 5) = cos (-2𝝅 / 5) and according to the periodicity cos (-2𝝅 / 5) = cos (-2𝝅 / 5 + 2𝝅), the angle at which the cosine has the same value as the angle 2𝝅 / 5 is

c) 𝝅-𝝅 / 2 = 𝝅 / 2, the angle and the supplement angle are the same. 𝝅 / 2 radians is 90 °. Here the value of the sine is 1 and between [0,2𝝅] the sine gets the value 1 only at this angle.

Example 6

Find the value of the cosine when we know that

We use Pythagorean trigonometric identity and substitute sine

The angle is between [𝝅/2, 𝝅], so the cosine is negative. So the answer is

Connection between sine and cosine

In the right triangle

This is true for all angles 𝛼. By radians

Tangent

The tangent of the oriented angle is sine divided by cosine. The divisor cannot be zero, so the tangent is not defined by angles with a cosine value of 0. The cosine is zero by the angles 𝝅 / 2 and -𝝅 / 2 and their multiples. The tangent is thus defined when

Period of the tangent is 𝝅

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