Trigonometry

Ontario 10 Academic (MPM2D)

Find angles in right-angled triangles

Lesson

We have already seen what the trigonometric ratios are:

We know wow to calculate with them, and how to find the length of unknown sides of right-angled triangles with them.

We can also use the trigonometric ratios to find the size of unknown angles. To do this we need any 2 of the side lengths.

Find the Size of an Unknown Angle

**label**the sides as O, A or H with respect to the position of the angle you want to find**identify**the appropriate trigonometric ratio that applies [either sine (sin), cosine (cos) or tangent (tan)]- using algebra,
**solve**the equation for the angle, (write the rule, fill in what you know, then solve using inverse operations) - reflect and
**check**(do a quick check on your calculator to confirm your answer is correct)

Find the angle indicated in this diagram.

1. label the sides as O, A or H with respect to the position of the angle

2. Identify the appropriate ratio that uses O and H. For this question it will be sine (sin)

3. Using algebra, solve the equation for the angle

(write the rule) $\sin\theta=\frac{O}{H}$`s``i``n``θ`=`O``H`

(fill in what you know) $\sin\theta=\frac{5}{8}$`s``i``n``θ`=58

(solve using inverse operations) $\theta=\sin^{-1}\left(\frac{5}{8}\right)$`θ`=`s``i``n`−1(58) use a calculator for this bit!

$\theta=38.68$`θ`=38.68°

We looked already at calculating angles from a value, here is a reminder.

Find $\theta$`θ` if $\sin\theta=0.65$`s``i``n``θ`=0.65 answer to $2$2 decimal places

This question is asking us what the angle is if the ratio of the opposite and hypotenuse is $0.65$0.65. To answer this question you use the inverse sin button on your scientific calculator. Often it looks a bit like this $\sin^{-1}$`s``i``n`−1

$\sin\theta=0.65$`s``i``n``θ`=0.65

$\sin^{-1}$`s``i``n`−1 $0.65=40.54$0.65=40.54°

Find the value of the angle indicated.

We have the opposite and adjacent sides here, so the ratio I will use is tangent (tan).

$\tan\theta=\frac{O}{A}$`t``a``n``θ`=`O``A` write the rule

$\tan\theta=\frac{14.77}{12.24}$`t``a``n``θ`=14.7712.24 fill in what we know

$\theta=\tan^{-1}\left(\frac{14.77}{12.24}\right)$`θ`=`t``a``n`−1(14.7712.24) use inverse operations to rearrange, and then use a calculator

$\theta=50.35$`θ`=50.35°

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

Consider the given figure.

Find the unknown angle $x$

`x`, correct to two decimal places.Find $y$

`y`, correct to two decimal places.Find $z$

`z`correct to two decimal places.

The person in the picture sights a pigeon above him. If the angle the person is looking at is $\theta$`θ`, find $\theta$`θ` in degrees.

Round your answer to two decimal places.

Determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem