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Finding Unknown Angles


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)


Question 1





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) $\sin\theta=\frac{O}{H}$sinθ=OH

(fill in what you know)  $\sin\theta=\frac{5}{8}$sinθ=58

(solve using inverse operations)   $\theta=\sin^{-1}\left(\frac{5}{8}\right)$θ=sin1(58)    use a calculator for this bit!



Using the inverse operation for sin/cos/tan 

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

Find $\theta$θ if  $\sin\theta=0.65$sinθ=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 $\sin^{-1}$sin1


$\sin^{-1}$sin1 $0.65=40.54$0.65=40.54°

Question 2

Find the value of the angle indicated.

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

$\tan\theta=\frac{O}{A}$tanθ=OA              write the rule

$\tan\theta=\frac{14.77}{12.24}$tanθ=14.7712.24        fill in what we know

$\theta=\tan^{-1}\left(\frac{14.77}{12.24}\right)$θ=tan1(14.7712.24)     use inverse operations to rearrange, and then use a calculator



Question 3

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

A right-angled triangle with vertices labeled A, B and C. Vertex A is at the top, B at the bottom right, and C at the bottom left. A small square at vertex A indicates that it is a right angle. Side interval(BC), which is the side opposite vertex A, is the hypotenuse and is marked with a length of 25. The angle located at vertex B is labelled x. Side interval(AB), descending from the right angle at vertex A to vertex B, is  marked with a length of 7, and is adjacent to the angle x. Side interval(AC) is opposite the angle x.

Question 4


Consider the given figure.

  1. Find the unknown angle $x$x, correct to two decimal places.

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

  3. Find $z$z correct to two decimal places.

Question 5

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.

The image depicts a right-angled triangle formed by dashed lines. A person in blue standing on the ground has a horizontal line of sight that represents the base of the triangle which measures 12m indicating the base length. From the base of the triangle, the vertical height of the pigeon above the man is 27m. The angle between the base and the hypotenuse is marked with a theta (θ) symbol, indicating an unknown angle to be potentially calculated.

  1. Round your answer to two decimal places.


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