Trigonometry-G11-Maths-T4P2 – Study Guide Docs

Trigonometric definitions

Trigonometry describes relationships between angles and sides. For a right-angled triangle, identify the side opposite the angle, the side adjacent to the angle and the hypotenuse.

\sin θ = \frac{opposite}{hypotenuse}

\cos θ = \frac{adjacent}{hypotenuse}

\tan θ = \frac{opposite}{adjacent}

The mnemonic SOH CAH TOA helps learners remember these definitions.

On the Cartesian plane, if the terminal arm of an angle passes through the point with coordinates shown below, the radius is calculated using the theorem of Pythagoras.

P (x; y)

r = \sqrt{x^{2} + y^{2}}

\sin θ = \frac{y}{r}

\cos θ = \frac{x}{r}

\tan θ = \frac{y}{x}

The signs depend on the quadrant:

First quadrant: all three ratios are positive.
Second quadrant: only sine is positive.
Third quadrant: only tangent is positive.
Fourth quadrant: only cosine is positive.

Special angles

The exact values of the special angles must be known.

\begin{matrix} θ & 0^{°} & 30^{°} & 45^{°} & 60^{°} & 90^{°} \\ \sin θ & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1 \\ \cos θ & 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & 0 \\ \tan θ & 0 & \frac{\sqrt{3}}{3} & 1 & \sqrt{3} & undefined \end{matrix}

Example

Evaluate the following without a calculator.

2 \sin 30^{°} + \cos 60^{°}

Substitute the exact values.

= 2 (\frac{1}{2}) + \frac{1}{2}

= \frac{3}{2}

Trigonometric identities

An identity is true for every value for which both sides are defined. The two fundamental identities are:

\tan θ = \frac{\sin θ}{\cos θ}

\sin^{2} θ + \cos^{2} θ = 1

Useful forms of the square identity include:

\sin^{2} θ = 1 - \cos^{2} θ

\cos^{2} θ = 1 - \sin^{2} θ

Example

Simplify the following expression.

\frac{1 - \sin^{2} θ}{\cos θ}

Use the square identity.

= \frac{\cos^{2} θ}{\cos θ}

= \cos θ

Exam Tip

When proving an identity, work with one side only, usually the more complicated side. Do not assume that the two sides are equal before proving the identity.

Reduction formulae

Reduction formulae express trigonometric ratios of large or negative angles in terms of acute angles.

\sin (90^{°} - θ) = \cos θ

\cos (90^{°} - θ) = \sin θ

\sin (180^{°} - θ) = \sin θ

\cos (180^{°} - θ) = - \cos θ

\tan (180^{°} - θ) = - \tan θ

\sin (180^{°} + θ) = - \sin θ

\cos (180^{°} + θ) = - \cos θ

\tan (180^{°} + θ) = \tan θ

\sin (360^{°} - θ) = - \sin θ

\cos (360^{°} - θ) = \cos θ

\tan (360^{°} - θ) = - \tan θ

For negative angles:

\sin (- θ) = - \sin θ

\cos (- θ) = \cos θ

\tan (- θ) = - \tan θ

Example

Simplify the following expression.

\sin 150^{°}

= \sin (180^{°} - 30^{°})

= \sin 30^{°}

= \frac{1}{2}

General solutions of trigonometric equations

A general solution gives all possible angles. The integer represents any whole number, including negative integers and zero.

For sine:

\sin θ = \sin α

θ = α + 360^{°} n

θ = 180^{°} - α + 360^{°} n, n \in Z

For cosine:

\cos θ = \cos α

θ = 360^{°} n \pm α, n \in Z

For tangent:

\tan θ = \tan α

θ = α + 180^{°} n, n \in Z

Trigonometric equations

To solve a trigonometric equation over a given interval, first isolate the trigonometric ratio, find the reference angle and then use the signs in the quadrants.

Example

Solve the equation over the given interval.

2 \sin θ - 1 = 0, 0^{°} \leq θ \leq 360^{°}

Isolate the sine ratio.

2 \sin θ = 1

\sin θ = \frac{1}{2}

Find the reference angle.

θ_{ref} = 30^{°}

Sine is positive in the first and second quadrants.

θ = 30^{°}

θ = 180^{°} - 30^{°}

θ = 150^{°}

Therefore:

θ = 30^{°} or 150^{°}

Common Mistake

Do not give only the reference angle. Check all quadrants and make sure every answer lies in the required interval.

Trigonometric graphs

The basic sine graph passes through the origin and has a wave shape.

y = \sin x

Its period is:

360^{°}

Its range is:

- 1 \leq y \leq 1

The cosine graph starts at its maximum value.

y = \cos x

Its period is also:

360^{°}

Its range is:

- 1 \leq y \leq 1

The tangent graph consists of separate increasing branches.

y = \tan x

Its period is:

180^{°}

It has vertical asymptotes at:

x = 90^{°} + 180^{°} n, n \in Z

Its range is all real numbers.

Transformations of trig graphs

A transformed sine or cosine graph can be written as:

y = a \sin b (x - p) + q

y = a \cos b (x - p) + q

A transformed tangent graph can be written as:

y = a \tan b (x - p) + q

The constants control the shape and position of the graph. When sketching, first identify the amplitude, period, horizontal shift, vertical shift and any reflection.

Amplitude

The amplitude is the vertical distance from the centre line to a maximum or minimum point. For sine and cosine graphs:

amplitude = | a |

Example

For the following graph, the amplitude is three.

y = 3 \sin x

The maximum and minimum values are:

y_{max} = 3

y_{min} = - 3

A tangent graph has no amplitude because it has no maximum or minimum value.

Period

The period is the horizontal length of one complete cycle.

For sine and cosine:

period = \frac{360^{°}}{| b |}

For tangent:

period = \frac{180^{°}}{| b |}

Example

Determine the period of the graph.

y = \cos 2 x

period = \frac{360^{°}}{2}

period = 180^{°}

Horizontal and vertical shifts

A horizontal shift moves a graph left or right. In the following form, the graph shifts by the value represented by the horizontal parameter.

y = \sin (x - p)

A positive value produces a shift to the right, while a negative value produces a shift to the left.

A vertical shift moves the graph upwards or downwards.

y = \sin x + q

The centre line is:

y = q

For sine and cosine, the range becomes:

q - | a | \leq y \leq q + | a |

Reflections

Multiplying the function by a negative number reflects its graph in the x-axis.

y = - \sin x

Replacing the input with its negative reflects the graph in the y-axis.

y = f (- x)

Sine and tangent are odd functions, while cosine is an even function.

\sin (- x) = - \sin x

\tan (- x) = - \tan x

\cos (- x) = \cos x

Sine rule

The sine rule is used when a side and its opposite angle are known. It is useful for ASA, AAS and certain SSA triangles.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Example

Calculate the unknown side.

A = 40^{°}, B = 70^{°}, a = 8 cm

\frac{b}{\sin 70^{°}} = \frac{8}{\sin 40^{°}}

b = \frac{8 \sin 70^{°}}{\sin 40^{°}}

b \approx 11,7 cm

Cosine rule

The cosine rule is used when three sides are involved or when two sides and the included angle are known.

a^{2} = b^{2} + c^{2} - 2 b c \cos A

To calculate an angle:

\cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}

Example

Calculate the third side of a triangle.

b = 7 cm, c = 10 cm, A = 60^{°}

a^{2} = 7^{2} + 10^{2} - 2 (7) (10) \cos 60^{°}

a^{2} = 49 + 100 - 70

a^{2} = 79

a = \sqrt{79}

a \approx 8,9 cm

Area rule

The area rule applies when two sides and the included angle are known.

Area = \frac{1}{2} b c \sin A

Equivalent forms are:

Area = \frac{1}{2} a c \sin B

Area = \frac{1}{2} a b \sin C

Example

Calculate the area of a triangle with two sides of eight centimetres and eleven centimetres and an included angle of fifty degrees.

Area = \frac{1}{2} (8) (11) \sin 50^{°}

Area \approx 33,7 {cm}^{2}

Solving 2D problems involving triangles

Draw and label a clear diagram. Mark all known sides and angles, and identify the required quantity.

Use the following strategy:

Choose SOH CAH TOA for right-angled triangles.
Choose the sine rule when an opposite side-angle pair is available.
Choose the cosine rule for three sides, or two sides and the included angle.
Choose the area rule for two sides and the included angle.
Use angles of elevation and depression carefully. They are measured from a horizontal line. Alternate angles between parallel horizontal lines are equal.
Give lengths to the required number of decimal places and include units. Keep unrounded calculator values during intermediate steps.

Interpretation of trigonometric graphs

Questions may require x-intercepts, y-intercepts, turning points, asymptotes, ranges or intervals where a graph is increasing, decreasing, positive or negative.

At an x-intercept:

y = 0

At the y-intercept:

x = 0

Intersections of two graphs represent solutions of an equation.

f (x) = g (x)

Maximum and minimum points can model quantities such as height, temperature or water level. The period represents the time taken for a repeating event to complete one cycle. A vertical shift gives the average or central value, while amplitude gives the maximum variation from that value.

Exam Tip

Read values from the scales, not from the apparent shape of the graph. State coordinates clearly, include units in contextual questions and use the required interval when giving solutions.

Study Guide For Grade

11

Mathematics

Trigonometric definitions

Special angles

Trigonometric identities

Reduction formulae

General solutions of trigonometric equations

Trigonometric equations

Trigonometric graphs

Transformations of trig graphs

Amplitude

Period

Horizontal and vertical shifts

Reflections

Sine rule

Cosine rule

Area rule

Solving 2D problems involving triangles

Interpretation of trigonometric graphs