Trigonometry - Topic 5

Question 1

Year 1 - Chapter 9.1

Equation for the cosine rule?

Answer 1

• a² = b² + c² - 2bcCos(A)
• cos(A) = (b² + c² - a²) / 2bc

Used to find a missing side if you know two sides and the angle between them

Question 2

Year 1 - Chapter 9.2

Equation for the sine rule?

Answer 2

• a/sinA = b/sinB = c/sinB
• sinA/a = sinB/b = sinC/c

Used to find the length of a missing side
sinθ = sin(180° - θ)

Question 3

Year 1 - Chapter 9.3

Equation for the area of a triangle?

Answer 3

A = 1/2(abSinC)

or

A = 1/2(base x height) for right-angle triangles

Question 4

Year 1 - Chapter 9.5

Describe the sine, cosine and tangent graphs

Answer 4

y = sinθ

• Repeats every 360° and crosses at the x-axis at 180°
• Has a minimum value of -1 and a maximum value of 1
• Peaks at y=1 at 90°, troughs at y=-1 at 270°

y = cosθ

• Repeats every 360° and crosses at the x-axis at 90°
• Has a minimum value of -1 and a maximum value of 1
• Peaks at y=1 at 0°, troughs at y=-1 at 180°

y = tanθ

• Repeats every 360° and crosses at the x-axis at 180°
• Has no maximum or minimum value
• Has vertical asymtotes at 90° and 270°

Question 5

Year 1 - Chapter 9.6

How do transformations affect trigonometric graphs?

Answer 5

• y = cos(θ - 90°) makes the graph move 90° to the right
• y = 2sin(θ) stretches the graph by a factor of 2, so now the maximum value is 2, and the minimum value is -2
• y = -tan(θ) makes the graph invert in the x-axis
• y = -1 + cos(θ) makes the graph translate down by -1, so now the maximum value is 0, and the minimum -2
• y = sin(2θ) makes the graph stretch horizontally by a scale factor of 1/2. In this case, the graph now intersects the x-axis at every 90°, and peaks at 45° instead of 90°
• y = tan(-θ) makes the graph invert in the y-axis

Question 6

Year 1 - Chapter 10.1

P(x,y) in a unit circle

Answer 6

cosθ = x = x-coordinate of P
sinθ = y = y-coordinate of P
tanθ = y/x = gradient (m) of OP

You can use these definitions to find the values of sine, cosine and tangent for any angle θ. You always measure positive angles θ anticlockwise from the positive x-axis.

Question 7

Year 1 - Chapter 10.1

How does the CAST diagram determine trigonometric ratios?

Answer 7

Top right quadrant (0° - 90°) = First quadrant, bottom right quadrant (270° - 360° or 0°) = Fourth quadrant.

• First quadrant - sinθ, cosθ and tanθ are all positive
• Second quadrant - sinθ is positive, cosθ and tanθ are negative
• Third quadrant - tanθ is positive, sinθ and cosθ are negative
• Fourth quadrant - cosθ is positive, sinθ and tanθ are negative

Question 8

Year 1 - Chapter 10.2

List of standard angles:

Answer 8

• sin30° = 1/2
• sin45° = √2/2
• sin60° = √3/2
• cos30° = √3/2
• cos45° = √2/2
• cos60° = 1/2
• tan30° = √3/3
• tan45° = 1
• tan60° = √3

Question 9

Year 1 - Chapter 10.3

List of trigonometric identities:

Answer 9

• For all values of θ, sin²θ + cos²θ ≡ 1
• For all values of θ such that cosθ ≠ 0, tanθ ≡ sinθ/cosθ

Question 10

Year 1 - Chapter 10.4

In simple trigonometric equations, where will solutions be within?

Answer 10

• Solutions to sinθ = k only exist when -1≤k≤1
• Solutions to cosθ = k only exist when -1≤k≤1
• Solution to tanθ = p exist for all values of p
• arccos in the range of 0≤θ≤180°
• arcsin in the range of -90°≤θ≤90°
• arctan in the range of -90°≤θ≤90°

Question 11

Year 1 - Chapter 10.5

In harder trigonometric equations, where will the solutions be?

Example

Answer 11

Solve the equation cos3θ = 0.766, in the interval 0≤θ≤360°.
↓
Let X = 3θ
So cosX° = 0.766
As X = 3θ
then as 0≤X≤3 x 360°
So the interval for X is 0≤X≤1080°
↓
X = 40°, 320°, 400°, 680°… 1040°
3θ = 40°, 320°, 400°, 680°… 1040°
So θ = 13.3°, 107°, 133°, 227°… 347°

Question 12

Year 1 - Chapter 10.6

Solve a two set solutions question for trigonometric quadratics

Example

Answer 12

5sin²x + 3sinx - 2 = 0
↓
(5sinx - 2)(sinx + 1) = 0
↓
5sinx - 2 = 0 and sinx + 1 = 0
↓
sinx = 2/5 or -1

Question 13

Year 2 - Chapter 5.1

List of standard radians:

Answer 13

• 2π radians = 360°
• π radians = 180°
• π/2 radians = 90°
• π/3 radians = 60°
• 1 radian = 180°/π ≈ 57.3°
• π/4 radians = 45°
• π/6 radians = 30°

Question 14

Year 2 - Chapter 5.1

List of standard trigonometric ratios:

In radians

Answer 14

• sin(π/6) = 1/2
• sin(π/3) = (√3)/2
• sin(π/4) = (√2)/2
• cos(π/6) = (√3)/2
• cos(π/3) = 1/2
• cos(π/4) = (√2)/2
• tan(π/6) = (√3)/3
• tan(π/3) = √3
• tan(π/4) = 1

Question 15

Year 2 - Chapter 5.1

List of CAST diagram acute angles:

In radians

Answer 15

• sin(π - θ) = sinθ
• sin(π + θ) = -sinθ
• sin(2π - θ) = -sinθ
• cos(π - θ) = -cosθ
• cos(π + θ) = -cosθ
• cos(2π - θ) = cosθ
• tan(π - θ) = -tanθ
• tan(π + θ) = tanθ
• tan(2π - θ) = -tanθ

Question 16

Year 2 - Chapter 5.2

Equation for arc length?

Answer 16

L = rθ

L is the length of the arc
r is the radius of the circle
θ is the angle, in radians

Question 17

Year 2 - Chapter 5.3

Equation for the area of a sector?

Answer 17

A = (1/2)(r²θ)

r is the radius of the circle
θ is the angle, in radians

Question 18

Year 2 - Chapter 5.3

Equation for the area of a segment?

Answer 18

A = (1/2)r²(θ - sinθ)

r is the radius of the circle
θ is the angle, in radians

Question 19

Year 2 - Chapter 5.5

What are the small angle approximations for sinθ, cosθ and tanθ?

Answer 19

When θ is small and measured in radians:

sinθ ≈ θ
tanθ ≈ θ
cosθ ≈ 1 - (θ²/2)