6.01 Exact ratios and the unit circle

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°=1√2​
• $\cos45^\circ=\frac{1}{\sqrt{2}}$cos45°=1√2​
• $\tan45^\circ=\frac{1}{1}$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.

 

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

Question 2

Question 3