1. Trigonometry – Without a Calculator – Demonstration
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2. Trigonometry Without a Calculator
SOH
CAH
TOA
Trigonometry Without a Calculator
What are the hidden
3 lengths and 1 angle?
30° 2
45° 2 2 1 3
45°
60° 1 1 2

3. SOH
CAH
TOA
Trigonometry Without a Calculator
What trigonometric values can we calculate?
30° 2
45° 2 2 1 3
45°
60° 1 1 2
sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
cos 60° = 1 2
tan 60° = = 3
tan 45° = 1 1 =1

4. 1 0° 1 Trigonometry Without a Calculator1
In a hypothetical triangle where one angle is 0°,
the hypotenuse is equal to the adjacent side.
sin 0° = 0 1 =0
cos 0° = 1 1 =1
tan 0° = 0 1 =0

5. sin 30° = 6 𝑥 = 1 2 𝑥 6 cm 6= 𝑥 2 30° 12=𝑥 DEMO sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
tan 60° = = 3
cos 60° = 1 2
tan 45° = 1 1 =1
sin 30° = 6 𝑥
= 1 2 𝑥
6 cm
6= 𝑥 2
30°
12=𝑥

6. cos 45° = 6 𝑥 = 1 2 𝑥 6= 𝑥 2 45° 6 cm 6 2 =𝑥 DEMO sin 45° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
tan 60° = = 3
cos 60° = 1 2
tan 45° = 1 1 =1
cos 45° = 6 𝑥
= 1 2 𝑥
6= 𝑥 2
45°
6 cm
6 2 =𝑥

7. denominator & simplify.
DEMO
sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
tan 60° = = 3
cos 60° = 1 2
tan 45° = 1 1 =1
tan 60° = 12 𝑥
= 3
12=𝑥 3
60° 𝑥 =𝑥
12 cm
Rationalise the
denominator & simplify.
4 3 =𝑥

8. denominator & simplify.
DEMO
sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
tan 60° = = 3
cos 60° = 1 2
tan 45° = 1 1 =1
= 1 2
sin 45° = 𝑥 4
4 cm 𝑥
𝑥= 4 2
45°
Rationalise the
denominator & simplify.
𝑥=2 2

9. denominator & simplify.
DEMO
sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
tan 60° = = 3
cos 60° = 1 2
tan 45° = 1 1 =1 =
sin 60° = 9 𝑥 𝑥
60°
9= 𝑥 3 2
18=𝑥 3
9 cm
Rationalise the
denominator & simplify. =𝑥
=6 3

10. Use the equilateral and isosceles triangles
to first find trigonometric values,
to then find the missing lengths.
Keep your answers in surd form. 2 2 1 𝒂 1 2
7 cm
30°
10 cm 𝒄
8 cm 𝒅
60°
45°
30°
6 cm 𝒃

11. 2 2 1 𝑎=14 cm 𝑎 1 2 7 cm 30° 10 cm 𝑐=8 2 cm 𝑐 8 cm 𝑑=2 3 cm 𝑑 60° 45°
Use the equilateral and isosceles triangles
to first find trigonometric values,
to then find the missing lengths.
Keep your answers in surd form. 2 2 1
𝑎=14 cm 𝑎 1 2
7 cm
30°
10 cm
𝑐=8 2 cm 𝑐
8 cm
𝑑=2 3 cm 𝑑
60°
45°
30°
6 cm
𝑏=5 cm 𝑏

12. 30° 2 45° 2 2 1 3 45° 60° 1 1 2 Trigonometry Without a Calculator
What trigonometric values can we calculate?
30° 2
45° 2 2 1 3
45°
60° 1 1 2
sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
cos 60° = 1 2
tan 60° = = 3
tan 45° = 1 1 =1

13. 30° 2 45° 2 2 1 3 45° 60° 1 1 2 Trigonometry Without a Calculator
What trigonometric values can we calculate?
30° 2
45° 2 2 1 3
45°
60° 1 1 2
sin 45° = 1 2
cos 30° =
sin 30° = 1 2
tan 30° =
cos 45° = 1 2
sin 60° =
cos 60° = 1 2
tan 60° = = 3
tan 45° = 1 1 =1

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