 # Find an Angle Using the Sin Ratio

In this worksheet, students will use the Sin ratio to find the missing angle of a triangle. Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Geometry and Measures, Mensuration

Curriculum subtopic:   Mensuration and Calculation, Triangle Mensuration

Difficulty level:   ### QUESTION 1 of 10 Question

Why is this person hanging upside down?

To look at something in a slightly different way.

We know that anything starting with the letters tri, means three things.

Three sides on a triangle, three wheels on a tricycle, three events in a triathlon and three ratios to learn in trigonometry.

Ah yes, our favourite trigonometry

(SOH, CAH, TOA)

You may have used the formula triangle to help find a missing side of a right angled triangle.

We can use  also them to help us find the missing angle in a right angled triangle.

Here we are going to look at the Sin formula triangle.

We will use it the same way as when finding a missing side, except for something slightly different at the end.

S is for SIN (which will be given as an angle)

O is for opposite side from the angle

H is the hypotenuse

The line in the middle means divide.

To use this triangle we cover up what it is we want to find (sin means angle) and we are left with a formula to follow.

The good old formula, where would we be without it. If we want to find the angle, The formula we are left with is opposite ÷ hypotenuse. Let's give it a go.

IMPORTANT NOTE: Make sure your calculator is set to degrees. You can tell as a D appears at the top of the screen.

Example 1 1. Label the triangle.

2. Find the two sides you want.  You want to find the angle, the only information to help us is the length of the hypotenuse H, and the length of the opposite side O.

3. Look at your formula triangle, cover up Sin (meaning angle)(see above)

4.  We are left with O ÷  H

5. In your calculator type in 3 ÷ 7 = 0.42. This is clearly not correct.

This is where the slight difference occurs.  You use the inverse sin button  (sin -1)on your calculator.  This is the upside down bit.

You usually access this by pressing shift and then sin

Example 2 1. Label the triangle

2. Find the two sides you want.  You want to find the angle, the only information to help us is the length of  H, and the length of O.

3. Look at your formula triangle, cover up Sin (meaning angle)(see above)

4.  We are left with O ÷  H

5. In your calculator type in 5 ÷ 8 = 0.625.

This is where the slight difference occurs.  You use the inverse sin button  (sin -1)on your calculator.

You usually access this by pressing shift and then sin

Over to you.. Find the angle in this triangle.

Answer has been rounded to 2 decimal places

41.81°

52.43°

38.21°

55.55° Find the size of the angle hypotenuse ÷ opposite opposite ÷ hypotenuse 42.42° 36.87° 40.12 ° The formula I need is The angle is hypotenuse ÷ opposite opposite ÷ hypotenuse 42.42° 36.87° 40.12 ° The formula I need is The angle is 34.98°

44.38°

64.28°

54.18°  62.73° 63.37° 64.46° The angle is 62.73° 63.37° 64.46° The angle is Find the size of the angle Answer has been rounded to 2 decimal places (2 d.p)

43.81°

45.32°

48.29°

47.62°

• Question 1 Find the angle in this triangle.

Answer has been rounded to 2 decimal places

41.81°
EDDIE SAYS
To find an angle, you are always going to need the opposite and the hypotenuse, but it can be easy to get the opposite and adjacent muddled up when there are measurements on all sides of the triangle. So care labelling is needed. Here you are given the opposite and hypotenuse only, so there is little room for error. 8 ÷ 12 = 0.66 0.66 sin-1 = 41.81 (to 2 d.p)
• Question 2 Find the size of the angle

56.4
EDDIE SAYS
opposite ÷ hypotenuse 5 ÷ 6 = 0.8333 0.8333 Sin-1 = 56.4 sin sational!!
• Question 3 hypotenuse ÷ opposite opposite ÷ hypotenuse 42.42° 36.87° 40.12 ° The formula I need is The angle is
EDDIE SAYS
Labelling the triangle correctly is key to unlocking he correct formula, particularly here when you have all three sides labelled. You know you need the opposite divided by the hypotenuse 9 ÷ 15 = 0.6 sin-1 0.6 = 36.87 to 2 decimal places
• Question 4 EDDIE SAYS
How are you getting on using the formula triangle? It is easier when finding the angle as the formula comes out the same each time. O ÷ H 18 ÷ 22 = 0.8181 sin-1 0.8181 = 54.90° sin tillating stuff huh?
• Question 5 54.18°
EDDIE SAYS
Are you getting used to this now? 15 ÷ 18.5 =0.8108 sin-1 0.8108 = 54.18°
• Question 6 43.81
EDDIE SAYS
Are you on a roll now. 4.5 ÷ 6.5 = 0.6923.. sin-1 0.6923... = 43.81° (to 2 d.p) Have you noticed that all angles come out as acute angles. If you have got anything over 90° then you have made a mistake somewhere.
• Question 7 62.73° 63.37° 64.46° The angle is
EDDIE SAYS
To be honest once you can label your triangle correctly you can leave the rest to your scientific calculator. What a great friend it is at a time like this. 8 ÷ 9 = 0.888 sin-1 0.888 = 62.73°
• Question 8 EDDIE SAYS
Calculators rock. The only thing you have to be careful of, is rounding to the correct number of decimal places. 10 ÷18 = 0.555 sin-1 0.555 = 33.75 (to 2 d.p)
• Question 9 Find the size of the angle

48.59
EDDIE SAYS
Whaaaat -this isn't a right angled triangle. What do I do now? You must have been waiting for the curved ball. Just turn it into one by drawing a line from the vertex to the opposite side. Don't forget to halve the base to give the correct length for the opposite side. Nearly there. I hope you haven't found this to tri-ing!! 4.5 ÷ 6 = 0.75 sin -1 0.75 = 48.59° (to 2 d.p)
• Question 10 Answer has been rounded to 2 decimal places (2 d.p)

43.81°
EDDIE SAYS
Are you ready to sin out now? One final check though 4.5 ÷ 6.5 = 0.6923... sin-1 0.6923 = 43.81(to 2 d.p)
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