 7.01 Exact ratios and the unit circle

Lesson

You'll have noticed by now that when you find angles using trigonometric ratios, you often get long decimal answers. If, for example, you put $\cos30^\circ$cos30° into the calculator, you will see an answer of $0.86602$0.86602... which we'd have to round. However, when you take cos, sin or tan of some angles, you can express the answer as an exact number, rather than a decimal. It just may include irrational numbers. We often use these exact ratios in relation to $30^\circ$30°, $45^\circ$45° and $60^\circ$60°.

Let's look at how to do this now.

Exact value triangles

$45$45 degree angles

Below is a right-angle isosceles triangle, with the equal sides of $1$1 unit. Using Pythagoras' theorem, we can work out that the hypotenuse is $\sqrt{1^2+1^2}=\sqrt{2}$12+12=2 units. The angles in a triangle add up to $180^\circ$180° and the base angles in an isosceles triangle are equal, so we can also work out that the two unknown angles are both $45^\circ$45°. We can then use our trig ratios to determine the exact values of the following:

• $\sin45^\circ=\frac{1}{\sqrt{2}}$sin45°=12
• $\cos45^\circ=\frac{1}{\sqrt{2}}$cos45°=12
• $\tan45^\circ=\frac{1}{1}$tan45°=11$=$=$1$1

$30$30 and $60$60 degree angles

To find the exact ratios of $30$30 and $60$60 degree angles, we need to start with an equilateral triangle with side lengths of $2$2 units. Remember all the angles in an equilateral triangle are $60^\circ$60°. Then we are going to draw a line that cuts the triangle in half into two congruent triangles. The base line is cut into two $1$1 unit pieces and cuts one of the $60^\circ$60° angles in half. Now let's just focus on one half of this triangle. We can calculate the length of the centre line to be $\sqrt{3}$3 using Pythagoras' theorem. We can then use our trig ratios to determine the exact values of the following:

• $\sin30^\circ=\frac{1}{2}$sin30°=12
• $\cos30^\circ=\frac{\sqrt{3}}{2}$cos30°=32

• $\sin60^\circ=\frac{\sqrt{3}}{2}$sin60°=32
• $\cos60^\circ=\frac{1}{2}$cos60°=12

Notice that the $\sin60^\circ=\cos30^\circ$sin60°=cos30°. It is true for any two complementary angles that $\sin x=\cos\left(90^\circ-x\right)$sinx=cos(90°x).

Observations

Now, an isosceles right-angled triangle may not have its sides measuring $1$1,$1$1 and $\sqrt{2}$2, but however large it is, it will always have two $45^\circ$45° angles and the ratios of the sides will always be the same as in the table. The same applies to the triangle with $60^\circ$60° and $30^\circ$30° angles. As long as a triangle is similar to one of these triangles (it has the same angles) we can use the exact values.

Exact values

We have found the exact values for the following using the $45$45 and $30$30 $60$60 triangles:

sin cos tan
$30^\circ$30° $\frac{1}{2}$12 $\frac{\sqrt{3}}{2}$32 $\frac{1}{\sqrt{3}}$13
$45^\circ$45° $\frac{1}{\sqrt{2}}$12 $\frac{1}{\sqrt{2}}$12 $1$1
$60^\circ$60° $\frac{\sqrt{3}}{2}$32 $\frac{1}{2}$12 $\sqrt{3}$3

We also have the exact values that don't describe physical triangles, at $0^\circ$0° and $90^\circ$90°:

sin cos tan
$0^\circ$0° $0$0 $1$1 $0$0
$90^\circ$90° $1$1 $0$0 undefined

Unit circle

The unit circle provides us with a visual understanding that the trigonometric functions of $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ exist for angles larger than what can be contained in a right-angled triangle.

The unit circle definitions of $\sin\theta$sinθ and $\cos\theta$cosθ tell us that the value of these functions will be the $x$x and $y$y-values respectively of a point on the unit circle after having rotated by an angle of measure $\theta$θ in the anticlockwise direction. Or, if $\theta$θ is negative, then the point is rotated in the clockwise direction. Definition of $\cos\theta$cosθ and $\sin\theta$sinθ can extend beyond $0^\circ\le\theta\le90^\circ$0°≤θ≤90°.

We can divide this into four quadrants as shown below: Consider the following definitions for each quadrant:

• Measures between $0^\circ$0° and $90^\circ$90° are said to be in the first quadrant.
• Measures between $90^\circ$90° and $180^\circ$180° are said to be in the second quadrant.
• Measures between $180^\circ$180° and $270^\circ$270° are said to be in the third quadrant.
• Measures between $270^\circ$270° and $360^\circ$360° are said to be in the fourth quadrant.

Practice questions

QUESTION 1

Use the exact value triangles in the diagram below to answer the following: 1. What is the exact value of $\cos45^\circ$cos45°?

2. What is the exact value of $\cos60^\circ$cos60°?

3. What is the exact value of $\cos30^\circ$cos30°?

Question 2

$\theta$θ is an angle in a right-angled triangle where $\tan\theta=\frac{1}{\sqrt{3}}$tanθ=13.

Solve for the exact value of $\theta$θ.

Question 3

Consider the point $\left(x,y\right)$(x,y) on the unit circle. 1. Use an equation to find the value of $x$x.

2. Use an equation to find the value of $y$y.

Outcomes

VCMMG368 (10a)

Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies.