1. Starter A triangular pyramid is made from baked bean tins
The top layer has 1 tin, the next layer 3 tins, next 6 tins and so on
How many tins are there altogether in 8 layers?

2. Calculate with the Primary Trigonometric Ratios
We are Learning to……
Calculate with the Primary Trigonometric Ratios

3. The tangent ratio the length of the opposite side
the length of the adjacent side
The ratio of
is the tangent ratio.
The value of the tangent ratio depends on the size of the angles in the triangle. θ O P S I T E
We say,
tan θ =
opposite
adjacent
A D J A C E N T

4. What is the value of tan 71°?
The tangent ratio
What is the value of tan 71°?
In a right-angled triangle with an angle of 71°, what is the ratio of the opposite side to the adjacent side?
This is the same as asking:
To work this out we can accurately draw a right-angled triangle with a 71° angle and measure the lengths of the opposite side and the adjacent side.

5. What is the value of tan 71°?
The tangent ratio
What is the value of tan 71°?
It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 71° are similar.
The length of the opposite side divided by the length of the adjacent side will always be the same value as long as the angle is the same.
In this triangle,
71°
11.6 cm
4 cm
tan 71° =
opposite
adjacent
Notice that the tangent ratio differs from the sine and cosine ratios in that it can be larger than 1. This is because when the angle we are concerned with is greater than 45°, the side opposite that angle will be longer than the side adjacent to the angle. When we divide a larger number by a smaller number the answer is always greater than 1. =
11.6 4
= 2.9

6. The tangent ratio using a calculator
What is the value of tan 71°?
To find the value of tan 71° using a scientific calculator, start by making sure that your calculator is set to work in degrees.
Key in:
tan 7 1 =
Some calculators require the size of the angle to be keyed in first, followed by the tan key.
Your calculator should display
This is 2.90 to 3 significant figures.

7. The sine ratio the length of the opposite side
the length of the hypotenuse
The ratio of
is the sine ratio.
The value of the sine ratio depends on the size of the angles in the triangle. θ O P S I T E H Y N U
We say:
sin θ =
opposite
hypotenuse
The sine ratio depends on the size of the opposite angle. We say that the sine of the angle is equal to the length of the opposite side divided by the length of the hypotenuse.
Sin is mathematical shorthand for sine. It is still pronounced as ‘sine’.

8. What is the value of sin 65°?
The sine ratio
What is the value of sin 65°?
In a right-angled triangle with an angle of 65°, what is the ratio of the opposite side to the hypotenuse?
This is the same as asking:
To work this out we can accurately draw a right-angled triangle with a 65° angle and measure the lengths of the opposite side and the hypotenuse.

9. What is the value of sin 65°?
The sine ratio
What is the value of sin 65°?
It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 65° are similar.
The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same.
In this triangle,
65°
10 cm
11 cm
sin 65° =
opposite
hypotenuse
This ratio can also be demonstrated using the similar right-angled activity on slide 7. = 10 11
= 0.91 (to 2 d.p.)

10. The sine ratio using a calculator
What is the value of sin 65°?
To find the value of sin 65° using a scientific calculator, start by making sure that your calculator is set to work in degrees.
Key in:
sin 6 5 =
Some calculators require the size of the angle to be keyed in first, followed by the sin key.
Your calculator should display
This is to 3 significant figures.

11. The cosine ratio the length of the adjacent side
the length of the hypotenuse
The ratio of
is the cosine ratio.
The value of the cosine ratio depends on the size of the angles in the triangle. θ
We say,
cos θ =
adjacent
hypotenuse
A D J A C E N T H Y P O T E N U S

12. What is the value of cos 53°?
The cosine ratio
What is the value of cos 53°?
In a right-angled triangle with an angle of 53°, what is the ratio of the adjacent side to the hypotenuse?
This is the same as asking:
To work this out we can accurately draw a right-angled triangle with a 53° angle and measure the lengths of the adjacent side and the hypotenuse.

13. What is the value of cos 53°?
The cosine ratio
What is the value of cos 53°?
It doesn’t matter how big the triangle is because all right-angled triangles with an angle of 53° are similar.
The length of the opposite side divided by the length of the hypotenuse will always be the same value as long as the angle is the same.
In this triangle,
53°
6 cm
10 cm
cos 53° =
adjacent
hypotenuse
This ratio can also be demonstrated using the similar right-angled activity on slide 7. = 6 10
= 0.6

14. The cosine ratio using a calculator
What is the value of cos 25°?
To find the value of cos 25° using a scientific calculator, start by making sure that your calculator is set to work in degrees.
Key in:
cos 2 5 =
Some calculators require the size of the angle to be keyed in first, followed by the cos key.
Your calculator should display
This is to 3 significant figures.

15. Finding side lengths
If we are given one side and one acute angle in a right-angled triangle we can use one of the three trigonometric ratios to find the lengths of other sides. For example,
Find x to 2 decimal places.
We are given the hypotenuse and we want to find the length of the side opposite the angle, so we use:
56° x
12 cm
sin θ =
opposite
hypotenuse
Discuss how we decide which trigonometric ratio to use.
sin 56° = x 12
x = 12 × sin 56°
= 9.95 cm

16. What is the distance between the base of the ladder and the wall?
Finding side lengths
A 5 m ladder is resting against a wall. It makes an angle of 70° with the ground.
What is the distance between the base of the ladder and the wall?
We are given the hypotenuse and we want to find the length of the side adjacent to the angle, so we use:
5 m
cos θ =
adjacent
hypotenuse
70° x 5 x
cos 70° =
x = 5 × cos 70°
= 1.71 m (to 2 d.p.)

17. The three trigonometric ratiosθ O P S I T E H Y N U
A D J A C E N T
Sin θ =
Opposite
Hypotenuse
S O H
Cos θ =
Adjacent
Hypotenuse
C A H
Tan θ =
Opposite
Adjacent
T O A
Stress to pupils that they must learn these three trigonometric ratios.
Pupils can remember these using SOHCAHTOA or they may wish to make up their own mnemonics using these letters.
Remember:
S O H
C A H
T O A

18. McGraw-Hill 12 Page 80 Q1 – 4 BLM 2.4 #s 1 – 2
To succeed at this lesson today you need to…
1. Identify the opposite
2. Identify the adjacent
3. Use a formula triangle to find the missing length
McGraw-Hill 12 Page 80 Q1 – 4 BLM 2.4 #s 1 – 2

19. The inverse of sin 30° 0.5 sin θ = 0.5, what is the value of θ?
To work this out use the sin–1 key on the calculator.
sin–1 0.5 =
30°
sin–1 is the inverse of sin. It is sometimes called arcsin.
sin
Make sure that pupils can locate the sin–1 key on their calculators.
Stress that sin and sin–1 are inverse functions.
sin 30° = 0.5 and sin–1 0.5 = 30°.
Remind pupils of the use of –1 to denote the multiplicative inverse or reciprocal. This is an extension of this notation.
sin–1
30°
0.5

20. The inverse of cos 60° 0.5 Cos θ = 0.5, what is the value of θ?
To work this out use the cos–1 key on the calculator.
cos–1 0.5 =
60°
Cos–1 is the inverse of cos. It is sometimes called arccos.
cos
Make sure that pupils can locate the cos–1 key on their calculators.
Stress that cos and cos–1 are inverse functions.
cos 60° = 0.5 and cos–1 0.5 = 60°.
cos–1
60°
0.5

21. The inverse of tan 45° 1 tan θ = 1, what is the value of θ?
To work this out use the tan–1 key on the calculator.
tan–1 1 =
45°
tan–1 is the inverse of tan. It is sometimes called arctan.
tan
Make sure that pupils can locate the tan–1 key on their calculators.
Stress that tan and tan–1 are inverse functions.
tan 45° = 1 and tan–1 1 = 45°.
tan–1
45° 1

22. Finding angles Find θ to 2 decimal places.
5 cm
8 cm
Find θ to 2 decimal places.
We are given the lengths of the sides opposite and adjacent to the angle, so we use:
tan θ =
opposite
adjacent
On the calculator we can key in tan–1 (8 ÷ 5). This avoids rounding errors when the ratio cannot be written exactly as a decimal.
tan θ = 8 5
θ = tan–1 (8 ÷ 5)
= 57.99° (to 2 d.p.)

23. Finding angles
Use this activity to practice finding the size of angles given two sides in a right-angled triangle.

24. Homework McGraw-Hill 12 Page 81 Q 6, 7, 9 & 10 BLM 2.4 #s 3 - 5
If you don’t have a scientific calculator…
EITHER:
(a) Buy one (the preferred option because you will need it for your exam) or
(b) Use the calculator on your phone. You will probably have to put the angle in first and then press tan/sin/cos.