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6.01 The trigonometric ratios

Lesson

Introduction

In right-angled triangles, the trigonometric functions can be used to construct simple relationships between the sides and angles of the triangle. These can be referred to as the trigonometric ratios.

Trigonometric ratios

A right angled triangle A B C with angle theta at A and right angle at B.

Consider the triangle with given angle \theta.

Since it is the longest side, as well as being opposite the right angle, we know that AC is the hypotenuse.

A right angled triangle A B C with angle theta at A and right angle at B. Ask your teacher for more information.

We can also see that the side BC is opposite the angle \theta, so we can refer to it as the opposite side. This leaves the side AB which is next to the angle \theta, so we can refer to it as the adjacent side. As such, with respect to the angle \theta, we can label the three sides.

Notice that if we instead choose \angle BCA to be \theta, the opposite and adjacent sides will switch to match the angle's new position.

Using the trigonometric functions and the given angle, we can express the ratios between the different pairs of sides as:\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}

If we are given one of the angles in a right-angled triangle and all of the side lengths, we can write the trigonometric ratios as fractions of the side lengths.

If we are only given two of the side lengths, we can calculate the third using Pythagoras' theorem.

For example:

A right angled triangle with an acute angle of theta, hypotenuse of 17 and adjacent side of 8.

This triangle has a given angle of \theta and two given side lengths.

Using Pythagoras' theorem, we can calculate the missing side length to be \sqrt{17^2-8^2}=15.

We can then identify all the sides by considering their position with respect to the angle \theta.

  • \text{Hypotenuse}= 17

  • \text{Opposite}= 15

  • \text{Adjacent}= 8

Using these values, we can then find the trigonometric ratios:

\sin \theta =\dfrac{\text{Opposite }}{\text{Hypotenuse }}= \dfrac{15}{17} \quad \quad \cos \theta =\dfrac{\text{Adjacent }}{\text{Hypotenuse }}= \dfrac{8}{17} \quad \quad \tan \theta =\dfrac{\text{Opposite }}{\text{Adjacent }}= \dfrac{15}{8}

As long as we are given one angle (that is not the right angle) and at least two side lengths, we can find the trigonometric ratios for any right-angled triangle.

Exploration

The applet below uses the mnemonic SOHCAHTOA in finding the trigonometric ratios. Move the slider to adjust the value of angle \theta.

Loading interactive...

For angle 0 \leq \theta \leq 90\degree: 0 \leq \sin \theta \leq 1, 0 \leq \cos \theta \leq 1, and \tan \theta \geq 0.

Examples

Example 1

Evaluate \sin \theta within \triangle ABC.

Triangle A B C has angle theta opposite of A C with length 24, angle alpha opposite of B C with length 7, and hypotenuse of 25.
Worked Solution
Create a strategy

Use the trigonometric ratio of sine.

Apply the idea

With respect to the given angle \theta, the opposite side is AC=24, and the hypotenuse is AB=25.

\displaystyle \sin \theta\displaystyle =\displaystyle \frac{\text{Opposite }}{\text{Hypotenuse }}Use the sine ratio
\displaystyle =\displaystyle \frac{24}{25}Substitute the values

Example 2

Find the value of \tan \theta in \triangle ABC.

Triangle A B C has angle theta opposite of A C with length x, angle alpha opposite of B C with length 7, and hypotenuse of 25.
Worked Solution
Create a strategy

Find the side length x using the Pythagoras' theorem c^2=a ^2+b ^2 and use the trigonometric ratio for tangent.

Apply the idea

\tan \theta = \dfrac{\text{Opposite }}{\text{Adjacent}} =\dfrac{x}{7}. So we need to use Pythagoras' Theorem to find the value of x.

\displaystyle c^2\displaystyle =\displaystyle a ^2+b ^2Use the Pythagoras' theorem
\displaystyle 25^2\displaystyle =\displaystyle 7 ^2+x ^2Substitute the values and x
\displaystyle x ^2\displaystyle =\displaystyle 25^2-7 ^2Subtract 7^2 from both sides
\displaystyle x ^2\displaystyle =\displaystyle 576Evaluate
\displaystyle x\displaystyle =\displaystyle \sqrt{576}Square root both sides
\displaystyle x\displaystyle =\displaystyle 24Evaluate

So, the length of the opposite side x is 24 and we can substitute this length into the trigonometric ratio for tangent.

\displaystyle \tan \theta\displaystyle =\displaystyle \frac{\text{Opposite }}{\text{Adjacent}}Use the tangent ratio
\displaystyle =\displaystyle \frac{24}{7}Substitute the values
Idea summary
A right angled triangle A B C with angle theta at A and right angle at B. Ask your teacher for more information.

For a right angled triangle, the trigonometric ratios are given by: \begin{aligned} \sin \theta &=\dfrac{\text{Opposite }}{\text{Hypotenuse }} \\ \cos \theta &=\dfrac{\text{Adjacent }}{\text{Hypotenuse }} \\ \tan \theta &=\dfrac{\text{Opposite }}{\text{Adjacent }} \end{aligned}

If we are only given two of the side lengths, we can calculate the third using Pythagoras' theorem: c^2=a^2+b^2.

Outcomes

VCMMG319

Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles.

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