7.01 Review of trigonometry for right-angled triangles

Lesson

Trigonometric ratios

A ratio is a statement of a mathematical relationship comparing two quantities, often represented as a fraction. If we consider an angle $\theta$θ in a right-angled triangle, we can construct various ratios to compare the lengths of the sides. In a right-angled triangle the ratios of the sides are the trigonometric ratios. Three common trigonometric ratios we use are Sine, Cosine and Tangent, we often shorten these names to sin, cos and tan respectively. They are given by the ratio of sides relative to the given angle $\theta$θ.

Trigonometric ratios
 $\sin\left(\theta\right)$sin(θ) $=$= $\frac{\text{Opposite}}{\text{Hypotenuse}}$OppositeHypotenuse​ $=$= $\frac{O}{H}$OH​ $\cos\left(\theta\right)$cos(θ) $=$= $\frac{\text{Adjacent}}{\text{Hypotenuse}}$AdjacentHypotenuse​ $=$= $\frac{A}{H}$AH​ $\tan\left(\theta\right)$tan(θ) $=$= $\frac{\text{Opposite}}{\text{Adjacent}}$OppositeAdjacent​ $=$= $\frac{O}{A}$OA​

Here is a picture of the above relationships, and for some people the mnemonic of SOHCAHTOA at the bottom is helpful to remember the sides that apply to the different ratios of sine, cosine and tangent.

From similar triangles we know that two triangles with the same angles will be similar and the ratio of corresponding sides will be equal. For a right-angled triangle with a given angle, say $20^\circ$20°, we know the third angle must be $70^\circ$70° and this would be true for any right-angled triangle with one angle of $20^\circ$20°. Hence, all right-angled triangles with an angle of $20^\circ$20° are similar and will have the same ratio of given sides. The calculator can approximate the ratio very accurately, typing in the calculator $\sin\left(20^\circ\right)$sin(20°), will tell us that the ratio for the opposite side divided by the hypotenuse for any right-angled triangle with an angle $20^\circ$20° is approximately $0.342$0.342.   We can use these trigonometric ratios to find unknown sides of a right-angled triangle given an angle or an unknown angle given two sides.

Practice questions

Question 1

Find the value of $f$f, correct to two decimal places.

Question 2

Find the value of $x$x to the nearest degree.

Special triangles

We can create right-angled triangles of varying side lengths and angle combinations. There are, however, 2 very special triangles that are referred to often in trigonometric studies.  These triangles are called exact value triangles, and they look like this.

Observations

- Any right-angled triangle with a $45^\circ$45° angle will be isosceles. This means it will have two equal sides (here we have the simplest case where they measure $1$1 unit each). How can we obtain the hypotenuse?

- The right-angled triangle with $30^\circ$30° and $60^\circ$60° angles can be obtained by cutting an equilateral triangle in half. See if you can start with an equilateral triangle of side length $2$2 to obtain the exact side lengths in the above triangle.

From these 2 triangles, we can construct trigonometric ratios for the angles 30, 45 and 60 degrees.

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.

These particular values are ones that we need to be familiar with for our continued study in high school trigonometry, as they will help us obtain exact rather than rounded values.

Practice questions

Question 3

Find the value of $\theta$θ if $\sin\theta=\frac{\sqrt{3}}{2}$sinθ=32, given that $0^\circ\le\theta\le90^\circ$0°θ90°.

Question 4

Given that $\tan\theta=\frac{1}{\sqrt{3}}$tanθ=13, find $\sin\theta$sinθ to two decimal places.

1. First, find the value of $\theta$θ, given that $0^\circ\le\theta\le90^\circ$0°θ90°.

2. Hence, find the value of $\sin\theta$sinθ.

Problem solving with right-angled triangles

When right angles are involved don't forget we can always use Pythagoras' theorem to find an unknown side given the other two sides. And in worded problems in trigonometry, the angle of elevation or depression is often used to describe the situation.

Useful facts

Pythagoras' Theorem:  $a^2+b^2=c^2$a2+b2=c2, where $c$c is the hypotenuse

Angle of Elevation:  the angle from the observer's horizontal line of sight looking UP at an object

Angle of Depression:  the angle from the observer's horizontal line of sight looking DOWN at an object

Practice questions

Question 5

A man standing at point $C$C, is looking at the top of a tree at point $A$A. Identify the angle of elevation in the figure given.

1. $\alpha$α

A

$\theta$θ

B

$\sigma$σ

C

$\alpha$α

A

$\theta$θ

B

$\sigma$σ

C

Question 6

From the top of a rocky ledge $188$188 m high, the angle of depression to a boat is $13^\circ$13°. If the boat is $d$d m from the foot of the cliff find $d$d correct to $2$2 decimal places.

Question 7

Consider the following diagram.

1. Find $y$y, correct to two decimal places.

2. Find $w$w, correct to two decimal places.

3. Hence, find $x$x, correct to one decimal place.

Outcomes

4.2.1.1

recall sine, cosine and tangent as ratios of side lengths in right-angled triangles