1. Measurment and Geometry
Best Practice #1
Trigonometry
AusVELS Level Students will identify similar triangles if the corresponding sides are in ratio and all corresponding angles equal.
Measurment and Geometry

2. Measurment and Geometry
Similar Triangles
Similarity can be used to find lengths or heights of large objects. It leads into problems involving trigonometry
Measurment and Geometry

3. Measurment and Geometry
Best Practice #2
Trigonometry
AusVELS Level 9.0
Students will identify the hypotenuse, adjacent and opposite sides with respect to the reference angle in a right-angled triangle
Measurment and Geometry

4. Labelling the right-angled triangle
Trigonometry is concerned with the connection between the sides and angles in any right angled triangle.
Angle
Measurment and Geometry

5. Measurment and Geometry
The sides of a right -angled triangle are given special names:
The hypotenuse, the opposite and the adjacent.
The hypotenuse is the longest side and is always opposite the right angle.
The opposite and adjacent sides refer to the reference angle, other than the 90o. A
Measurment and Geometry

6. Measurment and Geometry
Best Practice #3
Trigonometry
AusVELS Level 9.0
Students will define the Sine, Cosine and Tangent ratios for angles in right-angled triangles
Students will use Trigonometric notation eg. Sin A
Measurment and Geometry

7. Trigonometric Notation
There are three formulae involved in trigonometry:
Sin θ=
Cos θ=
Tan θ =
Measurment and Geometry

8. Measurment and Geometry
Add diagrams
S O H C A H T O A
Measurment and Geometry

9. Measurment and Geometry
Similar Triangles and Trig Ratios R B 5 20 3 12 C A 4 Q P 16
Let’s look at the 3 basic Trig ratios for these 2 triangles
They are similar triangles, since ratios of corresponding sides are the same. All similar triangles have the same trig ratios for corresponding angles.
Measurment and Geometry

10. Measurment and Geometry
Best Practice #4
Trigonometry
AusVELS Level 9.0
Students will select the appropriate trigonometric ratio to calculate unknown sides in a right-angled triangle, including using trigonometry to find an unknown side when the unknown letter is on the bottom of the fraction
Measurment and Geometry

11. Calculating an Unknown Side
To find a missing side from a right-angled triangle we need to know one angle and one other side.
Note: If
Cos45 =
To leave x on its own we need to move the ÷ 13.
It becomes a “times” when it moves.
x = 13 x Cos(45)
Measurment and Geometry

12. Measurment and Geometry
7 cm k
30o 1.
We have been given the adj and hyp so we use COSINE:
Cos A = H A
Cos A =
Cos 30 =
Pronumeral on other side
k = 7 x Cos 30
k = 6.1 cm
Measurment and Geometry

13. Measurment and Geometry
4 cm r
50o 2.
We have been given the opp and adj so we use TAN:
Tan A = A O
Tan A =
Tan 50 =
r = 4 x Tan 50
r = 4.8 cm
Measurment and Geometry

14. Measurment and Geometry
12 cm k
25o 3.
We have been given the opp and hyp so we use SINE:
Sin A = H O
sin A =
sin 25 =
k = 12 x Sin 25
k = 5.1 cm
Measurment and Geometry

15. Measurment and Geometry
There are occasions when the unknown letter is on the bottom of the fraction after substituting.
Cos45 =
Move the u term to the other side.
It becomes a “times” when it moves.
Cos45 x u = 13
To leave u on its own, move the cos 45 to other side, it becomes a divide.
u =
Measurment and Geometry

16. Measurment and Geometry
Best Practice #5
Trigonometry
AusVELS Level 9.0
To be able to use trigonometry to find an unknown side when the unknown letter is on the bottom of the fraction.
Measurment and Geometry

17. Calculating an Unknown Side- pronumeral on the bottom
When the unknown letter is on the bottom of the fraction we can simply swap it with the trig (sin A, cos A, or tan A) value.
Cos45 =
u =
Measurment and Geometry

18. Measurment and Geometryx
5 cm
30o 1.
Cos A = H
Cos 30 =
x = A
x = cm
sin A = m
8 cm
25o 2. H
sin 25 = O
m =
m = cm
Measurment and Geometry

19. Measurment and Geometry
Best Practice #6
Trigonometry
AusVELS Level 9.0
Students will select the appropriate trigonometric ratio to calculate unknown angles in a right-angled triangle
Measurment and Geometry

20. Calculating an Unknown Angle
To do this on the calculator, we use the sin-1, cos-1 and tan-1 function keys.
Measurment and Geometry

21. Measurment and Geometry
Example:
sin θ = find angle θ.
sin-1
0.1115 =
shift
sin ( )
θ = sin-1 (0.1115)
θ = 6.4o
2. cos θ = find angle θ
cos-1
0.8988 =
shift
cos ( )
θ = cos-1 (0.8988)
θ = 26° (to the nearest degree)
Measurment and Geometry

22. Measurment and Geometry
Finding an angle from a triangle
To find a missing angle from a right-angled triangle we need to know two of the sides of the triangle.
We can then choose the appropriate ratio, sin, cos or tan and use the calculator to identify the angle from the decimal value of the ratio.
14 cm
6 cm θ 1.
Find angle θ
Identify/label the names of the sides.
b) Choose the ratio that contains BOTH of the letters.
Measurment and Geometry

23. Measurment and Geometry
14 cm
6 cm θ 1.
We have been given the adjacent and hypotenuse so we use COSINE:
Cos θ = H A
Cos θ =
Cos θ =
Cos θ =
θ = cos-1 (0.4286)
θ = 64.6o
Measurment and Geometry

24. Measurment and Geometry
Find angle x 2.
8 cm
3 cm θ
Given adj and opp
need to use tan:
Tan θ = A O
Tan θ =
Tan θ =
Tan θ =
θ = tan-1 (2.6667)
θ = 69.4o
Measurment and Geometry

25. Measurment and Geometry3.
12 cm
10 cm θ
Given opp and hyp
need to use sin:
Sin θ =
sin θ =
sin θ =
sin θ =
θ = sin-1 (0.8333)
θ = 56.4o
Measurment and Geometry