Grade 11 Mathematics
Grade 11 · Term 3Mathematics

Trigonometry

We extend trig to include trig identities, the sine and cosine rules, area formula for a triangle, and solving trig equations in given intervals.

7.1 Trig Identities & Simplification7.2 Sine Rule, Cosine Rule & Area Formula
Week 1

7.1 Trig Identities & Simplification

  • Prove and apply the Pythagorean identity: $\sin^2\theta+\cos^2\theta=1$
  • Derive and apply quotient identity: $\tan\theta=\frac{\sin\theta}{\cos\theta}$
  • Simplify trig expressions using identities
🌍

Real-World Connection

Trig identities are like algebraic shortcuts for navigation computers — an aircraft's navigation system uses these identities millions of times per second to convert between angle representations and compute precise headings.

Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

Derived from Pythagoras on the unit circle; valid for all $\theta$

Derived identities

tan⁡2θ+1=sec⁡2θ1+cot⁡2θ=csc⁡2θ\tan^2\theta + 1 = \sec^2\theta \quad 1+\cot^2\theta = \csc^2\theta

Divide the Pythagorean identity by $\cos^2\theta$ or $\sin^2\theta$

Quotient Identity

tan⁡θ=sin⁡θcos⁡θ\tan\theta = \frac{\sin\theta}{\cos\theta}

$\cos\theta\neq0$

💡 Tip

When simplifying, choose the more complex side and work toward the simpler side. Key strategy: convert everything to sin⁡θ\sin\theta and cos⁡θ\cos\theta first, then use the Pythagorean identity.

Worked Examples

Worked Example

Prove a trig identity

Problem

Prove:sin⁡2θ1−cos⁡θ=1+cos⁡θProve: \dfrac{\sin^2\theta}{1-\cos\theta} = 1+\cos\theta
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
If sin⁡θ=0.6\sin\theta=0.6, find cos⁡2θ\cos^2\theta.
2
L1 · Knowledge1 mark
Express tan⁡θ\tan\theta in terms of sin⁡θ\sin\theta and cos⁡θ\cos\theta.
3
L2 · Routine Procedures3 marks
Simplify cos⁡2θ−1sin⁡θ\dfrac{\cos^2\theta-1}{\sin\theta}.
4
L2 · Routine Procedures3 marks
Prove sin⁡θtan⁡θ=cos⁡θ\dfrac{\sin\theta}{\tan\theta}=\cos\theta.
5
L3 · Complex Procedures4 marks
Simplify (sin⁡θ+cos⁡θ)2−1(\sin\theta+\cos\theta)^2-1.
6
L3 · Complex Procedures4 marks
Prove tan⁡θ+1tan⁡θ=1sin⁡θcos⁡θ\tan\theta+\dfrac{1}{\tan\theta}=\dfrac{1}{\sin\theta\cos\theta}.
7
L4 · Problem Solving4 marks
If sin⁡α+cos⁡α=62\sin\alpha+\cos\alpha=\frac{\sqrt{6}}{2}, find sin⁡αcos⁡α\sin\alpha\cos\alpha.
8
L4 · Problem Solving5 marks
Prove 11−sin⁡θ+11+sin⁡θ=2sec⁡2θ\dfrac{1}{1-\sin\theta}+\dfrac{1}{1+\sin\theta}=2\sec^2\theta.
Week 3

7.2 Sine Rule, Cosine Rule & Area Formula

  • Apply the sine rule: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
  • Apply the cosine rule: $a^2=b^2+c^2-2bc\cos A$
  • Apply the area formula: $\text{Area}=\frac{1}{2}ab\sin C$
  • Determine when to use each rule
🌍

Real-World Connection

Surveyors measuring land that has no right angle use the sine and cosine rules daily. Knowing two sides and the included angle (SAS) or two angles and a side (AAS), they calculate the third side without ever physically measuring it.

Sine Rule

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

Use when: AAS, ASA, SSA (ambiguous case)

Cosine Rule

a2=b2+c2−2bccos⁡Aa^2 = b^2+c^2-2bc\cos A

Use when: SAS (two sides + included angle) or SSS (three sides)

Area of Triangle

Area=12absin⁡C\text{Area} = \frac{1}{2}ab\sin C

Two sides $a$, $b$ and included angle $C$

💡 Tip

Quick guide: if you have an angle opposite a known side, try the sine rule first. If you have two sides and the INCLUDED angle (the angle BETWEEN them), use the cosine rule.

Worked Examples

Worked Example

Apply the cosine rule

Problem

In △ABC\triangle ABC: a=7a=7, b=5b=5, C=60°C=60°. Find side cc and the area.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge3 marks
In △ABC\triangle ABC, A^=40°\hat{A}=40°, B^=75°\hat{B}=75°, a=8a=8. Find bb using the sine rule.
2
L1 · Knowledge3 marks
Find the area of a triangle with sides 6 and 9 and included angle 50°.
3
L2 · Routine Procedures4 marks
In △PQR\triangle PQR: p=10p=10, q=12q=12, r=15r=15. Find P^\hat{P} (angle at PP).
4
L2 · Routine Procedures3 marks
Find side aa given b=7b=7, c=9c=9, A=120°A=120° using the cosine rule.
5
L3 · Complex Procedures4 marks
A triangle has area 24 cm² and two sides of 8 cm and 9 cm. Find the included angle.
6
L3 · Complex Procedures5 marks
Two ships leave a port. Ship A travels 50 km on bearing 040° and ship B travels 80 km on bearing 160°. Find the distance between them.
7
L4 · Problem Solving5 marks
In △ABC\triangle ABC, B^=60°\hat{B}=60°, a=5a=5, b=43b=4\sqrt{3}. Show there is only one possible triangle.
8
L4 · Problem Solving5 marks
Prove that the area of a regular hexagon with side aa is 332a2\dfrac{3\sqrt{3}}{2}a^2.

Next

Euclidean Geometry

Trigonometry Grade 11 Maths CAPS Notes & Examples | MathSciBuddy