NZ Level 6 (NZC) Level 1 (NCEA)
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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.

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.

  1. Round your answer to two decimal places.




Use trigonometric ratios and Pythagoras’ theorem in two and three dimensions


Apply geometric reasoning in solving problems


Apply right-angled triangles in solving measurement problems

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