Grade 12 Maths · Trigonometry · Solve a trig equation using identities

Solve a trig equation using identities Grade 12

Rewrite a trig equation with a double angle, compound angle or square identity so it has one ratio and one angle, factorise it, then give the general solution or the answers in an interval (for example 2x = x becomes (2 x - 1)( x + 1) = 0).

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General solution grade 12 TrigonometryKevinmathscience

Lesson 7: How To Find General Solutions of Trig Equations | Grade 12 TrigonometryMlungisi Nkosi Maths & Science

Solving Trigonometric Equations By Factoring & By Using Double Angle IdentitiesThe Organic Chemistry Tutor

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Question 7.4 · November 2024 · Paper 2 · National (DBE) · 4 marks

In the diagram below, the graph of

f (x) = tan x

is drawn for the interval

x \in [- 180°; 180°]

. $Diagram for this question$

7.4 Use the graphs to determine the general solution of

2 cos^{3} x - sin x = 0

.(4)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: solve a trig equation using identities

Work each one out first, then open the memo and check every tick.

3. Question 5.5.1 · May/June 2023 · Paper 2 · National (DBE) · 4 marks
5.5 Given: $3 tan 4 x = - 2 cos 4 x$
5.5.1 Without using a calculator, show that $sin 4 x = - 0, 5$ is the only solution to the above equation.(4)
Show the memo (4 ticks)Memo
1. ✓
  Identity $3 tan 4 x = - 2 cos 4 x$
  $3 (\frac{s i n 4 x}{c o s 4 x}) = - 2 cos 4 x$
  1 mark · Write tan as sin over cos
2. ✓
  1 − sin² 4x $3 sin 4 x + 2 cos^{2} 4 x = 0$
  $3 sin 4 x + 2 (1 - sin^{2} 4 x) = 0$
  1 mark · Solve a trig equation using identities
3. ✓
  Standard form $- 2 sin^{2} 4 x + 3 sin 4 x + 2 = 0$
  $2 sin^{2} 4 x - 3 sin 4 x - 2 = 0$
  1 mark · Write a quadratic in standard form
4. ✓
  Factors $(2 sin 4 x + 1) (sin 4 x - 2) = 0$
  $sin 4 x = - \frac{1}{2}$ or $sin 4 x \neq = 2$
  1 mark · Factorise trinomials with a leading coefficient
4. Question 5.3.2 · May/June 2024 · Paper 2 · National (DBE) · 5 marks
5.3 Given the expression: $\frac{s i n 150° + c o s ^{2} x - 1}{2}$
5.3.2 Hence, determine the general solution of $\frac{s i n 150° + c o s ^{2} x - 1}{2} = \frac{1}{25}$ (5)
Show the memo (5 ticks)Memo
1. ✓
  METHOD 1: Answer of 5.3.1 = 1/25 $\frac{c o s 2 x}{4} = \frac{1}{25}$
  $cos 2 x = \frac{4}{25}$
  1 mark · Solve a trig equation using identities
2. ✓
  METHOD 1: 2x = 80,79°ref $∠ = 80, 79...°$
  $2 x = 80, 79...° + k .360°$
  1 mark · Calculate an angle from a ratio
3. ✓
  METHOD 1: 2x = 279,20°or $2 x = 279, 20...° + k .360°$
  1 mark · Write the general solution
4. ✓
  METHOD 1: x = 40,40° and x = 139,60° $x = 40, 40°$ or $x = 139, 60°$
  1 mark · Write the general solution
5. ✓
  METHOD 1: + k.180°; k ∈ Z $x = 40, 40° + k .180°$ or $x = 139, 60° + k .180°$ ; $k \in Z$
  1 mark · Write the general solution
5. Question 6.2 · November 2024 · Paper 2 · National (DBE) · 6 marks
6.2 Given: $f (x) = 6 sin^{2} x - 11 cos (90° + x) + 7$

Solve for $x$ in the interval $x \in (0°; 360°)$ if $f (x) = 2$ .(6)
Show the memo (6 ticks)Memo
1. ✓
  Squaring both sides $6 sin^{2} x - 11 cos (90° + x) + 7 = 2$
  $6 sin^{2} x - 11 cos (90° + x) + 7 = 4$
  1 mark · Solve a surd equation
2. ✓
  $cos (90° + x) = - sin x$ $6 sin^{2} x - 11 (- sin x) + 7 = 4$
  $6 sin^{2} x + 11 sin x + 3 = 0$
  1 mark · Solve a trig equation using identities
3. ✓
  Factors $(3 sin x + 1) (2 sin x + 3) = 0$
  1 mark · Factorise trinomials with a leading coefficient
4. ✓
  Both equations $sin x = - \frac{1}{3}$ OR $sin x = - \frac{3}{2}$ (no solution)
  1 mark · Solve a quadratic by factorising
5. ✓
  Answer: 199,47°ref $∠ = 19, 47°$
  $x = 180° + 19, 47° = 199, 47°$
  1 mark · Find solutions in an interval
6. ✓
  Answer: 340,53° $x = 360° - 19, 47° = 340, 53°$
  1 mark · Find solutions in an interval
6. Question 5.3.2 · November 2023 · Paper 2 · National (DBE) · 5 marks
5.3 Given: $cos (A - B) = cos A cos B + sin A sin B$
5.3.2 Hence, or otherwise, determine the general solution of the equation $sin 48° cos x - cos 48° sin x = cos 2 x$ (5)
Show the memo (5 ticks)Memo
1. ✓
  Compound angle $sin 48° cos x - cos 48° sin x = cos 2 x$
  $sin (48° - x) = cos 2 x$
  1 mark · Solve a trig equation using identities
2. ✓
  Co-ratio $sin (48° - x) = sin (90° - 2 x)$
  1 mark · Use co-ratio identities
3. ✓
  Both equations $48° - x = 90° - 2 x + k .360°$ or
  $48° - x = 180° - (90° - 2 x) + k .360°$
  1 mark · Write the general solution
4. ✓
  General solution $x = 42° + k .360°$
  1 mark · Write the general solution
5. ✓
  General solution; $k \in Z$ $- 3 x = 42° + k .360°$
  $x = - 14° - k .120°$ ; $k \in Z$
  1 mark · Write the general solution

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