1. Area, Sine Rule, Cosine Rule – Outcomes
Use trigonometry to calculate the area of a triangle.
Solve problems using the sine rule (2D).
Solve problems using the cosine rule (2D).
Define sin 𝜃 and cos 𝜃 for all values of 𝜃.
Work with trigonometric ratios in surd form.

2. Find Areas using Trigonometry
Previously, we had used 1 2 ×𝑏𝑎𝑠𝑒×⊥ℎ𝑒𝑖𝑔ℎ𝑡 to calculate the area of triangles.
This can be difficult, particularly for non-right-angled triangles, so trigonometry offers us a broader option:
on page 16 of F&T booklet
Formula: 𝐴𝑟𝑒𝑎= 1 2 𝑎𝑏 sin 𝐶
𝑎, 𝑏, and 𝐶 can be exchanged for other angles and sides, provided that angle is between the two sides.
i.e. 𝐴𝑟𝑒𝑎= 1 2 𝑏𝑐 sin 𝐴 = 1 2 𝑎𝑐 sin 𝐵

3. Find Areas using Trigonometry
e.g. Find the areas of each of the following triangles:

4. Find Areas using Trigonometry
e.g. Evaluate the labelled variables in the following:

5. Find Areas using Trigonometry
2007 OL P2 Q5
Calculate the area of the triangle shown.
Give your answer correct to one decimal place.
𝐴= 1 2 𝑎𝑏 sin 𝐶
2010 OL P2 Q5
In the triangle 𝐴𝐵𝐶, 𝐴𝐵 = 6 𝑐𝑚, 𝐵𝐶 =5 𝑐𝑚 and ∠𝐴𝐵𝐶 = 135 𝑜 .
Calculate the area of the triangle correct to the nearest square centimetre.

6. Find Areas using Trigonometry
2014 OL P2 Q8
A wind turbine, used to generate electricity, has three equally spaced blades 65 metres long.
Write down the size of the angle between two blades.
Find the area of the triangle formed by joining the tips of the three blades, correct to the nearest whole number.
Gráinne stood at a point 𝐵, which is on level ground 100 metres from the base of the turbine. From there, she measured the angle of elevation to the top of the tower to be 60 𝑜 . Find the height of the tower.
𝐴= 1 2 𝑎𝑏 sin 𝐶

7. Find Areas using Trigonometry
2014 OL P2 Q8
[continued]
Gráinne recognises that her measurements may not be totally accurate. She read elsewhere that the actual height of the tower is 154 m.
If Gráinne measured the 100 m accurately, find the actual size of angle 𝐵, correct to the nearest degree.
Find the percentage error in Gráinne’s measurement of the angle of elevation, correct to one decimal place.
𝐴= 1 2 𝑎𝑏 sin 𝐶

8. Solve Problems using Sine Rule
𝑎 sin 𝐴 = 𝑏 sin 𝐵
e.g. Find 𝑎 in the following triangle:
𝑎 sin 32 = 20 sin 40
⇒𝑎= 20 sin 32 sin 40
⇒𝑎=16.5

9. Solve Problems using Sine Rule
𝑎 sin 𝐴 = 𝑏 sin 𝐵
Evaluate 𝑎, 𝑏, 𝑐, and 𝑑:

10. Solve Problems using Sine Rule
𝑎 sin 𝐴 = 𝑏 sin 𝐵
Evaluate 𝐴, 𝐵, and 𝐶:

11. Solve Problems using Sine Rule
2011 OL P2 Q5
Use the sine rule to calculate the value of 𝑥 in the diagram.
Give your answer correct to the nearest integer.
𝑎 sin 𝐴 = 𝑏 sin 𝐵
2006 OL P2 Q5
In the triangle Δ𝑎𝑏𝑐, 𝑎𝑏 =18.4, 𝑏𝑐 =14, and ∠𝑐𝑎𝑏 = 44 𝑜 .
Find |∠𝑏𝑐𝑎|, correct to the nearest degree.
Find the area of the triangle Δ𝑎𝑏𝑐, correct to the nearest whole number.

12. Solve Problems using Cosine Rule
𝑎 2 = 𝑏 2 + 𝑐 2
−2𝑏𝑐 cos 𝐴
e.g. Sketch the triangle Δ𝐴𝐵𝐶, where 𝐴𝐵 =12, 𝐴𝐶 =10, and ∠𝐴 = 30 𝑜 .
Find 𝑎.
𝑎 2 = − cos 30
⇒ 𝑎 2 = −
⇒ 𝑎 2 =
⇒𝑎=6.01

13. Solve Problems using Cosine Rule
𝑎 2 = 𝑏 2 + 𝑐 2
−2𝑏𝑐 cos 𝐴
Sketch each of the following triangles before finding the noted side or angle:
Δ𝐴𝐵𝐶, with ∠𝐴 = 70 𝑜 , 𝑏=4, 𝑐=9. Find 𝑎.
Δ𝑋𝑌𝑍, with ∠𝑌 = 112 𝑜 , 𝑥=2, 𝑦=3. Find 𝑧.
Δ𝑅𝑆𝑇, with 𝑅𝑆 =2, 𝑆𝑇 =3, 𝑅𝑇 =5. Find |∠𝑅|.
Δ𝐾𝐿𝑀, with 𝐾𝐿 =5, 𝐿𝑀 =10, 𝐾𝑀 =7. Find |∠𝐿|.
Δ𝐽𝐻𝐾, with ∠𝐻 = 130 𝑜 , 𝐽𝐻 =13, 𝐻𝐾 =8. Find |∠𝐾|.
Δ𝐷𝐸𝐹, with 𝑑=4, 𝑒=5, 𝑓=7. Find ∠𝐸 .
Δ𝑃𝑄𝑅, with 𝑝=50, 𝑞=70, 𝑟=60. Find the largest angle.
Adapted from FHSST Grade 11

14. Solve Problems using Cosine Rule
2011 OL P2 Q5
𝑃𝑄𝑅𝑆 is a quadrilateral with diagonal [𝑆𝑄]. 𝑅𝑆 =62, 𝑆𝑄 =35, ∠𝑆𝑄𝑅 = 82 𝑜 , ∠𝑆𝑃𝑄 = 60 𝑜 , 𝑆𝑃 =3𝑥, and 𝑃𝑄 =8𝑥.
𝑎 2 = 𝑏 2 + 𝑐 2
−2𝑏𝑐 cos 𝐴
Find |∠𝑄𝑅𝑆|, correct to the nearest degree, given that 0 𝑜 ≤ ∠𝑄𝑅𝑆 ≤ 90 𝑜 .
Find the value of 𝑥.

15. Solve Problems using Cosine Rule
2009 OL P2 Q5
A harbour is 6 km due East of a lighthouse. A boat is 4 km from the lighthouse.
The bearing of the boat from the lighthouse is 𝑁 40 𝑜 𝑊.
𝑎 2 = 𝑏 2 + 𝑐 2
−2𝑏𝑐 cos 𝐴
How far is the boat from the harbour?
Find the bearing of the boat from the harbour.

16. Solve Problems using Cosine Rule
2006 OL P2 Q5
The lengths of the sides of the triangle Δ𝑝𝑞𝑟 are 𝑝𝑞 =20, 𝑞𝑟 =14 and 𝑝𝑟 =12.
𝑎 2 = 𝑏 2 + 𝑐 2
−2𝑏𝑐 cos 𝐴
Find |∠𝑟𝑝𝑞|, correct to one decimal place.
Find |𝑟𝑡|, where 𝑟𝑡⊥𝑝𝑞. Give your answer correct to the nearest whole number.

17. Define sin 𝜃 and cos 𝜃 for all 𝜃
While we usually define sin 𝜃 and cos 𝜃 for angles between 0 and 90 degrees, some applications make it useful to define them for any angle.
e.g. Find 𝐴 in the triangle below to the nearest degree.

18. Define sin 𝜃 and cos 𝜃 for all 𝜃
To generalise the trig functions, we redefine them based on the unit circle.
The unit circle is a circle on the coordinate plane with centre 0, 0 and radius 1.

19. Define sin 𝜃 and cos 𝜃 for all 𝜃
Measuring the angle anti-clockwise from the positive 𝑥-axis:
cos 𝐴 is the 𝑥-coordinate where the radius meets the circumference,
sin 𝐴 is the 𝑦-coordinate where the radius meets the circumference.

20. Define sin 𝜃 and cos 𝜃 for all 𝜃
Using the unit circle, complete this table.
𝑨 (degrees) 90
180
270
360
sin 𝐴
cos 𝐴
Recall that angles measure anti-clockwise from the positive 𝑥-axis.
cos 𝐴 is the 𝑥-coordinate
sin 𝐴 is the 𝑦-coordinate

21. Define sin 𝜃 and cos 𝜃 for all 𝜃
Draw a unit circle including axes.
Draw radiuses at 60o, 120o, 240o, and 300o.
Find cos 𝐴 for each of the angles drawn.
Find sin 𝐴 for each of the angles drawn.

22. Define sin 𝜃 and cos 𝜃 for all 𝜃
Each angle can be reframed as the smallest angle made with the 𝑥- axis, called the reference angle.
For 60 𝑜 , 120 𝑜 , 240 𝑜 , and 𝑜 , this angle is 60 𝑜 .
Since cos 60 =0.5, cosine of each of the other angles is either 0.5 or −0.5 depending on what side of the circle it’s on.

23. Define sin 𝜃 and cos 𝜃 for all 𝜃
Recall that cos 𝐴 represents the 𝑥- coordinate of the circumference.
Thus, it is positive in the first and fourth quadrants of the circle.
Similarly, it is negative in the second and third quadrants of the circle.

24. Define sin 𝜃 and cos 𝜃 for all 𝜃
Similarly, sin 60 =0.866, so the sine of each of the other angles is either or −0.866.

25. Define sin 𝜃 and cos 𝜃 for all 𝜃
Recall that sin 𝐴 represents the 𝑦- coordinate of the circumference.
Thus, it is positive in the first and second quadrants of the circle.
Similarly, it is negative in the third and fourth quadrants of the circle.

26. Define sin 𝜃 and cos 𝜃 for all 𝜃
e.g. Solve each of the following for 0 ∘ ≤𝐴< 360 ∘ , correct to the nearest degree:
sin 𝐴 =0.5
cos 𝐴 =0.5
sin 𝐴 =0.3
cos 𝐴 =0.2
sin 𝐴 =−0.6
cos 𝐴 =−0.7
sin 𝐴 =−0.1
cos 𝐴 =−0.8

27. Work with Trig Ratios in Surd Form
Recall surds are square roots of whole numbers
For special angles 30 ∘ , 45 ∘ and 60 ∘ , the F&T booklet gives exact answers in surd form.
CASIO calculators will usually display these by default.
Try sin −
30°
45°
60°
𝐴 (degrees)
3 2
1 2
1 2
cos 𝐴
sin 𝐴
1 3 1 3
tan 𝐴

28. Work with Trig Ratios in Surd Form
If cos 𝐴 = , find two values of 𝐴 if 0 𝑜 ≤𝐴≤ 360 𝑜 .
If sin 𝐵 =− 1 2 , find two values of 𝐵 if 0 𝑜 ≤𝐵≤ 360 𝑜 .
If tan 𝐶 = 3 , find two values of 𝐶 if 0 𝑜 ≤𝐶≤ 360 𝑜 .

29. Work with Trig Ratios in Surd Form
2006 HL P2 Q4
Find the value of 𝜃 for which cos 𝜃 =− , 0 𝑜 ≤𝜃≤ 180 𝑜
2006 HL P2 Q4
Write down the values of 𝐴 for which
cos 𝐴 = 1 2 , where 0 𝑜 ≤𝐴≤ 360 𝑜
1998 HL P2 Q4
Find the values of 𝜃 for which
cos 𝜃 = , where 0 𝑜 ≤𝜃≤ 360 𝑜
2011 HL P2 Q5
Find the values of 𝑥 for which
3 tan 𝑥 = 3 , where 0 𝑜 ≤𝑥≤ 360 𝑜