Core Chapter 8: Non-Right Angled Triangle Trigonometry

Question 1

Describe the unit circle

Answer 1

Unit Circle:
- Circle with centre at origin (0,0)
- Radius of 1 unit
- Equation x²+y² =1

• Positive angles (θ) are measured anti-clockwise from the positive x-axis
• Negative angles (-θ) are measured clockwise from the positive x-axis
• x-coordinate of unit circle: cosθ
  At point (a,b),
  cosθ = adj/hyp = a/1 = a (x-coordinate)
• y-coordinate of unit circle: sinθ
  sinθ = opp/hyp = b/1 = b (y-coordinate)
• Domain of unit circle:
  -1≤x≤1 –> -1≤cosθ≤1
• Range of unit circle:
  -1≤y≤1 –> -1≤sinθ≤1
• Since the equation of the unit circle is x²+y² =1 –> x² = cos²θ and y² = sin²θ –>
  cos²θ + sin²θ = 1

Question 2

Use the unit circle to understand the rules of supplementary angles

Answer 2

Supplementary angles: when the sum of 2 angles add up to 180°
- Eg. The supplement of θ on the unit circle would be (180°-θ)
- The coordinates of P’ when coordinates of P (cosθ, sinθ) is ( cos (180°-θ), sin (180°-θ) )

Since the supplement of θ is in the negative x-axis and the positive y-axis:
cos (180°-θ) = -cosθ
sin (180°-θ) = sinθ

Question 3

Apply and prove the formula for area of a triangle (when triangle is not a right-angled triangle and the perpendicular height h is unknown)

Answer 3

Area of a triangle = 1/2 x base x height
Area of a triangle = 1/2 x a x b x sinC

Proof:
sinC = opp/hyp = h/b
h = b (sin C)
Area of a triangle
= 1/2 x a x h
= 1/2 x a x b (sin C)
= 1/2 x a x b x sinC

Question 4

State and apply the cosine rule to find:
- Unknown side of a triangle (when given 2 sides and an angle)
- Unknown angle of a triangle (when given 3 sides)

Answer 4

c² = a²+b²-2abcosC
(finding an unknown side when given 2 sides and an angle)

cosC = (a²+b²-c²)/2ab
(finding an unknown angle when given all 3 sides)

Question 5

State and apply the sine rule to find:
- Unknown side (when given 1 length and 2 corresponding angles)
- Unknown angle (When given 2 lengths and 1 corresponding angle)

State the ambiguous case of the sine rule

Answer 5

a/sinA = b/sinB = c/sinC

• If angles are found using the sine rule and there is insufficient information to determine the shape of the triangle, θ can be either the acute angle or its supplement obtuse angle (ambiguous case)