1. Trigonometry – Tangent – Angles – Demonstration
This resource provides animated demonstrations of the mathematical method.
Check animations and delete slides not needed for your class.

2. θ Hypotenuse Opposite Adjacent A right-angled triangle has 4 parts.
θ = Theta is either angle.
Hypotenuse
Opposite θ
Adjacent
Hypotenuse – always opposite the right-angle & always longest.
Opposite – always opposite θ.
Adjacent – next to θ.

3. TOA 𝑥 (O) 12 cm 9 cm (A) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 12 9
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
(O)
We want to find Tan θ.
(so cover Tan θ)
12 cm
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 12 9 𝑥
𝑇𝑎𝑛 − 𝑥=
=53.13°
9 cm
(A)

4. TOA 𝑥 (O) (A) 9 cm 7 cm 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 9 7
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
(O)
(A)
9 cm
7 cm
We want to find Tan θ.
(so cover Tan θ) 𝑥
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 9 7
𝑇𝑎𝑛 − 𝑥=
=52.13°

5. TOA 𝑥 5 cm 9 cm (O) (A) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 5 9
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A 𝑥
We want to find Tan θ.
(so cover Tan θ)
5 cm
9 cm
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 5 9
(O)
(A)
𝑇𝑎𝑛 − 𝑥=
=29.05°

6. TOA 𝑥 5 cm (O) (A) 8 cm 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 5 8
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
5 cm
(O)
(A)
We want to find Tan θ.
(so cover Tan θ)
8 cm
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 5 8 𝑥
𝑇𝑎𝑛 − 𝑥=
=32.01°

7. TOA 𝑥 (O) 12 cm 5 cm (A) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 12 5
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
(O)
12 cm
We want to find Tan θ.
(so cover Tan θ)
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 12 5 𝑥
5 cm
(A)
𝑇𝑎𝑛 − 𝑥=
=67.38°

8. TOA 𝑥 (O) 4 cm 8 cm (A) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 4 8
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
(O)
4 cm
We want to find Tan θ.
(so cover Tan θ) 𝑥
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 4 8
8 cm
(A)
𝑇𝑎𝑛 − 𝑥=
=26.57°

9. TOA 𝑥 (A) 10 cm 11 cm (O) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 11 10
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
(A)
10 cm 𝑥
We want to find Tan θ.
(so cover Tan θ)
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 11 10
11 cm
(O)
𝑇𝑎𝑛 − 𝑥=
=47.73°

10. TOA 𝑥 (A) 13 cm 8 cm (O) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 8 13
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A 𝑥
(A)
We want to find Tan θ.
(so cover Tan θ)
13 cm
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 8 13
𝑇𝑎𝑛 − 𝑥=
=31.61°
8 cm
(O)

11. TOA 𝑥 (A) 11 cm 20 cm (O) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 20 11
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
(A)
11 cm 𝑥
We want to find Tan θ.
(so cover Tan θ)
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 20 11
20 cm
(O)
𝑇𝑎𝑛 − 𝑥=
=61.19°

12. TOA 𝑥 14 cm (O) 3 cm (A) 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A 𝑇𝑎𝑛 θ= 𝑂 𝐴 𝑇𝑎𝑛 𝑥= 14 3
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Find the value of 𝑥 to 2dp. O
Tan θ A
We want to find Tan θ.
(so cover Tan θ)
14 cm
𝑇𝑎𝑛 θ= 𝑂 𝐴
𝑇𝑎𝑛 𝑥= 14 3 𝑥
(O)
𝑇𝑎𝑛 −
3 cm 𝑥=
=77.91°
(A)

13. TOA 𝑥 𝑥 𝑥 7 cm 12 cm 5 cm 3 cm 4 cm 5 cm 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O A
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Calculate 𝑥 for these three triangles. (2dp) O
Tan θ A
7 cm 𝑥
12 cm 𝑥
5 cm
3 cm
4 cm 𝑥
5 cm

14. TOA 𝑥 𝑥 𝑥 7 cm 12 cm 5 cm 3 cm 4 cm 5 cm 𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗 O 𝑥=54.46° A
Label the sides.
Write the formula.
Substitute & calculate.
𝑇𝑎𝑛 θ= 𝑂𝑝𝑝 𝐴𝑑𝑗
Calculate 𝑥 for these three triangles. (2dp) O
Tan θ A
𝑥=54.46°
7 cm 𝑥
12 cm 𝑥
5 cm
𝑥=30.96°
3 cm
4 cm 𝑥
𝑥=18.43°
5 cm

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