TRIGONOMETRY QUESTIONS AND ANSWERS GRADE 12

EXPLORING TRIGONOMETRY QUESTIONS AND ANSWERS GRADE 12

Activity 1

1. ∆ MNP is a right-angled triangle. Write down the trig ratios for:
   1. sin α
   2. sin β (4)
   3. tan β
2. cos α (3)
3. If MP = 13 and NP = 5, calculate cos β.
   
   [7]

Activity 2

1. If sin θ is negative and cos θ is positive, then which statement is true?
   1. 0° < θ < 90°
   2. 90° < θ < 180°
   3. 180° < θ < 270°
   4. 270° < θ < 360° (1)
2. If tan θ < 0 and cos θ < 0, then which statement is true?
   1. 0° < θ < 90°
   2. 90° < θ < 180°
   3. 180° < θ < 270°
   4. 270° < θ < 360° (1)
3. Will the following trig ratios be positive or negative?
   1. sin 315°
   2. cos (–215°)
   3. tan 215°
   4. cos 390° (4)
      [6]

Activity 3
If cos β = ^p/_√5 where p < 0 and β ∈ [180°; 360°], determine, using a diagram, an expression in terms of p for:

1. tan β
2. 2 cos²β – 1
   [6]

Activity 4
Without using a calculator, determine the value of:

1. cos 150°
2. sin (–45°)
3. tan 480°
   [7]

Activity 5
Write the trig ratios as the trig ratios of their co-functions:

1. sin 50°
2. cos 70°
3. sin 100°
4. cos 140°
   [4]

Summary
Any angle (obtuse or reflex) can be reduced to an acute angle by using:

• Convert negative angles to positive angles
• Reduce angles greater than 360°
• Use reduction formulae
• Use co-functions

Activity 6
Simplify without using a calculator:

1. sin(180° + x). cos 330°.tan 150° (4)
   sin x
2.       cos 750°.tan 315°.cos(–θ)      (8)
   cos(360°- θ).sin300°.sin(180°- θ)
3. tan 480°.sin 300°.cos 14°.sin(–135°) (9)
   sin104°.cos225°
4.     cos 260°.cos 170°   (7)
   sin10°.sin190°.cos350°
   [28]

Activity 7
Simplify the following expressions.

1.      cos (180°–x) sin (x –90°) – 1    (8)
   tan²(540° + x) sin(90°+x)cos(–x)
2. [sin(–θ) + cos(360° + θ)][cos(θ – 90°) + cos(180°+θ)] (3)
3. cos2θ (1 + tan2θ) (3)
4. 1 – cos²θ
   1– sin²θ (3)
   [17]

Activity 8
Prove the following identities:

sin x ∙tan x + cos x =   1
cos x (4)

(sin x + tan x) (   sin x   ) = sin x. tan x (7)
1 + cos x

1     =   cos x   + tan x (6)
cos x    1 + sin x

1   + tan x = tanx  (5)
tanx                sin²x
[22]

Hints for solving trig identities:

• Choose either the lefthand side or the righthand side and simplify it to look like the other side.
• If both sides look difficult, you can try to simplify on both sides until you reach a point where both sides are the same.
• It is usually helpful to write tan θ as sinθ
  cosθ .
• Sometimes you need to simplify sinθ  to tan θ.
  cos θ
• If you have sin²x or cos²x with +1 or –1, use the squares identities (sin²θ + cos²θ = 1).
• Find a common denominator when fractions are added or subtracted.
• Factorise if necessary – specify with examples i.e. common factor, DOPS, Trinomial, sum/diff of two cubes

Activity 9

1. If cos 20° = p, determine the following ratios in terms of p:
   1. cos 380°
   2. sin 110°
   3. sin 200° (6)
2. Determine the general solution for x in the following equations:
   1. 5 sin x = cos 320° (correct to 2 decimal places)
   2. 3 tan x + √3 = 0 (without using a calculator)
   3. tan x–1 = –3 (correct to one decimal place) (10)
      2
3. Determine x for x ∈[–180°; 180°] if 2 + cos (2x – 10°) = 2,537 (6)
   [22]

Activity 10
Do NOT use a calculator to answer this question. Show ALL calculations. Prove that:

1. cos 75° =√2 ( √3 –1) (5)
   4
2. Prove that cos(90° – 2x).tan(180° + x) + sin²(360° – x) = 3sin²x (7)
3. Prove that (tan x – 1)(sin 2x – 2cos²x) = 2(1 – 2sin x cos x) (7)
   [19]

Activity 11
Determine the general solution for x in the following:

1. sin 2x. cos 10° – cos 2x. sin 10° = cos 3x (8)
2. cos² x = 3 sin 2x (11)
3. 2sinx = sin(x + 30°) (5)
   [24]