Grade 10 Mathematics
Grade 10 · Term 3Mathematics

Trigonometric Functions & 2D Problems

We plot y = sinθ, y = cosθ and y = tanθ point-by-point, read off their key features, then study how the parameters a and q transform each graph. We also continue solving 2D right-angled triangle problems.

6.1 Basic Trig Graphs: sin, cos, tan6.2 Transformed Trig Graphs: Effect of a and q6.3 Two-Dimensional Trigonometry Problems
Week 1

6.1 Basic Trig Graphs: sin, cos, tan

  • Plot y = sinθ, y = cosθ and y = tanθ point-by-point for θ ∈ [0°; 360°]
  • Identify period, amplitude, domain, range and asymptotes for each graph
  • Read intercepts and turning points from the graphs
🌍

Real-World Connection

Sound waves and ocean waves are modelled by sinusoidal functions. The amplitude is the loudness (or wave height) and the period is how often the pattern repeats. Radio engineers tune circuits by controlling these exact features.

Key features of y = sinθ on [0°; 360°]

  • Amplitude: 1 (max value = 1, min value = −1)
  • Period: 360°
  • y-intercept: (0°; 0); zeros: 0°, 180°, 360°
  • Maximum: (90°; 1); Minimum: (270°; −1)
  • Domain: θ ∈ [0°; 360°]; Range: y ∈ [−1; 1]

Key features of y = cosθ on [0°; 360°]

  • Amplitude: 1 (max value = 1, min value = −1)
  • Period: 360°
  • y-intercept: (0°; 1) = maximum; zeros: 90°, 270°
  • Minimum: (180°; −1); Maximum also at (360°; 1)
  • Domain: θ ∈ [0°; 360°]; Range: y ∈ [−1; 1]

Key features of y = tanθ on [0°; 360°]

  • Vertical asymptotes at θ = 90° and θ = 270°
  • Period: 180°
  • Zeros at θ = 0°, 180°, 360°
  • Domain: θ ∈ [0°; 360°], θ ≠ 90°, θ ≠ 270°; Range: y ∈ ℝ
  • No amplitude (unbounded); passes through (45°; 1) and (225°; 1)

💡 Tip

Cosine is a shifted sine: y = cosθ = sin(θ + 90°). Their graphs are identical in shape — cosine just starts at its maximum while sine starts at zero.

Worked Examples

Worked Example

Plot y = sinθ point-by-point

Problem

Complete the table of values for y=sin⁡θy=\sin\theta for θ∈{0°; 30°; 45°; 60°; 90°; 120°; 150°; 180°; 210°; 240°; 270°; 300°; 330°; 360°}\theta\in\{0°;\,30°;\,45°;\,60°;\,90°;\,120°;\,150°;\,180°;\,210°;\,240°;\,270°;\,300°;\,330°;\,360°\} and identify the key features.

Worked Example

Read off all key features of y = sinθ

Problem

Fory=sin⁡θon[0°;360°],state:amplitude,period,range,x−intercepts,turningpoints.For y = \sin\theta on [0°;360°], state: amplitude, period, range, x-intercepts, turning points.

Worked Example

Compare sin and cos at key angles

Problem

Usinggraphs,findallθ∈[0°;360°]wheresin⁡θ=cos⁡θ.Using graphs, find all θ ∈ [0°; 360°] where \sin\theta = \cos\theta.

Worked Example

Behaviour of tanθ near asymptotes

Problem

Explainwhyy=tan⁡θhasasymptotesat90°and270°,andstatetheperiod.Explain why y = \tan\theta has asymptotes at 90° and 270°, and state the period.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge1 mark
State the range of y=cos⁡θy = \cos\theta.
2
L1 · Knowledge1 mark
At what values of θ ∈ [0°; 360°] does y = sinθ reach its maximum?
3
L2 · Routine Procedures3 marks
For y=tan⁡θy = \tan\theta, state the asymptotes on [0°; 360°] and the period.
4
L2 · Routine Procedures2 marks
Compare the y-intercepts of y=sin⁡θy = \sin\theta and y=cos⁡θy = \cos\theta.
5
L3 · Complex Procedures3 marks
On the graph of y=cos⁡θy = \cos\theta, how many zeros are there on (0°; 360°)? State them.
6
L3 · Complex Procedures3 marks
Describe the symmetry of y=sin⁡θy = \sin\theta about θ=90°\theta = 90°.
7
L4 · Problem Solving4 marks
For which values of θ ∈ [0°; 360°] is sin⁡θ>cos⁡θ\sin\theta > \cos\theta?
8
L4 · Problem Solving4 marks
Explain why y=sin⁡θy = \sin\theta and y=tan⁡θy = \tan\theta both pass through (180°;0)(180°; 0) but their behaviour around that point is very different.
Week 2

6.2 Transformed Trig Graphs: Effect of a and q

  • Study the effect of a on y = a·sinθ + q (amplitude = |a|; reflection if a < 0)
  • Study the effect of q (vertical shift; new range [q − |a|; q + |a|])
  • Sketch y = a·sinθ + q, y = a·cosθ + q, y = a·tanθ + q; find equations from given graphs
🌍

Real-World Connection

Adjusting the volume on a speaker changes the amplitude (parameter a) — the wave gets taller or shorter. Adding a DC offset to an electrical signal shifts the whole wave up or down (parameter q). Engineers control both independently.

Property / Rule

Effect of a on y = a·f(θ)

The parameter a stretches or compresses the graph vertically. If a > 0 the shape is unchanged; if a < 0 the graph is reflected about the x-axis. The new amplitude is |a| for sin and cos.

Amplitude of y=asin⁡θ=∣a∣\text{Amplitude of }y=a\sin\theta = |a|

Property / Rule

Effect of q on y = f(θ) + q

Adding q shifts the entire graph q units vertically. The asymptote of y = a·tanθ + q moves to y = q. The range of y = a·sinθ + q becomes [q − |a|; q + |a|].

Range of y=asin⁡θ+q:  [q−∣a∣  ;  q+∣a∣]\text{Range of }y=a\sin\theta+q:\;[q-|a|\;;\;q+|a|]

Finding the equation from a graph

a=max−min2,q=max+min2a = \frac{\text{max} - \text{min}}{2},\quad q = \frac{\text{max} + \text{min}}{2}

max = maximum y-value of the graph; min = minimum y-value

💡 Tip

To find the equation from a sketch: (1) Check if it is sin, cos or tan by the shape. (2) Calculate a and q using the max and min. (3) Check whether it is reflected (a negative) by looking at whether the graph starts going up or down.

Worked Examples

Worked Example

Sketch y = 2sinθ − 1

Problem

Sketchy=2sin⁡θ−1forθ∈[0°;360°],showingallkeypoints.Sketch y = 2\sin\theta - 1 for \theta \in [0°;360°], showing all key points.

Worked Example

Find the equation from a graph

Problem

Agraphshapedlikeacosinecurvehasmaximum4andminimum−2.Themaximumisatθ=0°.Findtheequation.A graph shaped like a cosine curve has maximum 4 and minimum −2. The maximum is at \theta = 0°. Find the equation.

Worked Example

Sketch y = −cosθ + 2

Problem

Sketchy=−cos⁡θ+2,statingallkeyfeatures.Sketch y = -\cos\theta + 2, stating all key features.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
State the amplitude and range of y=3sin⁡θy = 3\sin\theta.
2
L1 · Knowledge2 marks
State the vertical shift and the equation of the asymptote of y=tan⁡θ+4y = \tan\theta + 4.
3
L2 · Routine Procedures3 marks
For y=−2cos⁡θ+1y = -2\cos\theta + 1: state the maximum, minimum and range.
4
L2 · Routine Procedures3 marks
A sin graph has maximum 5 and minimum 1. Find a and q.
5
L3 · Complex Procedures5 marks
Sketch y=−sin⁡θ+2y = -\sin\theta + 2 on [0°; 360°], showing all intercepts and turning points.
6
L3 · Complex Procedures4 marks
A cosine graph has a minimum of −3 at θ = 180° and passes through (0°; 1). Find the equation.
7
L4 · Problem Solving4 marks
For f(θ)=2sin⁡θ−1f(\theta)=2\sin\theta-1 and g(θ)=−1g(\theta)=-1, find all values of θ ∈ [0°; 360°] where f(θ)=g(θ)f(\theta)=g(\theta).
8
L4 · Problem Solving5 marks
The graph of y=acos⁡θ+qy=a\cos\theta+q passes through (90°; −1) and (180°; 3). Find a, q and the range.
Week 3

6.3 Two-Dimensional Trigonometry Problems

  • Solve 2D problems involving right-angled triangles using sin, cos, tan
  • Apply angles of elevation and depression
  • Solve multi-step problems combining two right-angled triangles
🌍

Real-World Connection

Surveyors measure distances they cannot walk across — rivers, mountains, buildings — by setting up two triangles from a measured baseline. Each triangle uses one trig ratio, and the answer connects across both. This technique, called triangulation, built every map before GPS.

Definition

Angle of Elevation

The angle measured upward from the horizontal to a line of sight looking toward a higher object.

tan⁡(elevation)=heighthorizontal distance\tan(\text{elevation}) = \frac{\text{height}}{\text{horizontal distance}}

Definition

Angle of Depression

The angle measured downward from the horizontal to a line of sight looking toward a lower object. Angle of depression from A = angle of elevation to A (alternate angles with horizontal).

depression from top=elevation from bottom (alternate ∠s)\text{depression from top} = \text{elevation from bottom (alternate } \angle\text{s)}

💡 Tip

Always draw a diagram first. Label the right angle, the known sides/angles, and the unknown. Choose the ratio (sin, cos, or tan) that connects two knowns and one unknown.

Worked Examples

Worked Example

Angle of elevation — find height

Problem

From a point 40 m from the base of a tower, the angle of elevation to the top is 62°. Find the height of the tower.

Worked Example

Angle of depression — find distance

Problem

From the top of a lighthouse 80 m tall, the angle of depression to a boat is 35°. How far is the boat from the base?

Worked Example

Two-triangle problem

Problem

Points A and B are on level ground. C is directly above B. From A, the angle of elevation of C is 40° and AB = 30 m. D is between A and B, DA = 12 m. Find the angle of elevation of C from D.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 · Knowledge2 Q
L2 · Routine Procedures2 Q
L3 · Complex Procedures2 Q
L4 · Problem Solving2 Q
1
L1 · Knowledge2 marks
A ladder 8 m long leans against a wall, making a 70° angle with the ground. How high up the wall does it reach?
2
L1 · Knowledge2 marks
A ramp rises 3 m over a horizontal distance of 10 m. Find the angle of inclination.
3
L2 · Routine Procedures3 marks
From a point 50 m from a building, the angle of elevation to the roof is 28°. Find the height of the building.
4
L2 · Routine Procedures3 marks
An observer at the top of a 60 m cliff sees a ship at an angle of depression of 22°. How far is the ship from the base of the cliff?
5
L3 · Complex Procedures5 marks
From point A on level ground, the angle of elevation of the top T of a vertical pole is 36°. From point B, 25 m closer to the pole on the same line, the angle is 54°. Find the height of the pole.
6
L3 · Complex Procedures4 marks
In right triangle PQR with Q^=90°\hat{Q}=90°, PQ = 5 and PR = 13. Find sin⁡P\sin P, cos⁡P\cos P and the area of the triangle.
7
L4 · Problem Solving5 marks
A person stands between two vertical poles of heights 12 m and 18 m. The angles of elevation to the tops are 40° and 50° respectively. Find the distance between the poles.
8
L4 · Problem Solving4 marks
Show that in a right-angled triangle with acute angle θ, sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1, and use this to find cosθ if sin⁡θ=513\sin\theta = \frac{5}{13}.

Next

Analytical Geometry

Trigonometric Functions & 2D Problems Grade 10 Maths CAPS Notes & Examples | MathSciBuddy