Unit notes and practice questions

Grade 10 Mathematics unit 4: Trigonometric functions

Everything here is for unit 4 of the Grade 10 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

6
Cards: 10
Practice questions on unit 4: 186–236
The textbook pages for unit 4

What unit 4 covers

1.Radian Measure of angle
2.Basic Trigonometric Function
3.Trigonometric Identities & Equation
4.Application

Short notes for unit 4

The cards filed under this chapter on the Grade 10 Mathematics page. Open one at a time.

Arc Length FormulaFormula

l = r \theta

l = r \theta

$l$: arc length $\text{m}$
$r$: radius of the circle $\text{m}$
$\theta$: central angle in radians $\text{rad}$

When you use it

Use this formula to find the length of an arc subtended by a central angle in a circle of a given radius.

Watch out

The central angle must be converted to radians before calculating. Using degrees directly will give an incorrect answer.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Degree and Radian Confusion in Arc CalculationsCommon mistake

When you use it

Check this when solving problems involving arc length, circular motion, or angle conversion.

Watch out

Never insert angle values in degrees into the formula l = r theta without multiplying by pi / 180 first.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Double Angle Sine FormulaFormula

\sin 2\theta = 2\sin\theta\cos\theta

\sin 2\theta = 2\sin\theta\cos\theta

$\theta$: angle $\text{rad}$

When you use it

Use this formula to calculate the sine of twice an angle using the sine and cosine of the single angle.

Watch out

Sine of two theta is not equal to two times sine of theta. You must multiply two by both sine theta and cosine theta.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Pythagorean Trigonometric IdentityFormula

\sin^2\theta + \cos^2\theta = 1

\sin^2\theta + \cos^2\theta = 1

$\theta$: angle $\text{rad}$

When you use it

Use this identity to find the value of sine given cosine or cosine given sine for any angle.

Watch out

Remember to determine the correct positive or negative sign of the resulting trigonometric value based on the quadrant where the angle terminates.

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Reference Angles by QuadrantReference tableQuadrant I · Quadrant II · Quadrant III · +1

Quadrant	Angle Range	Reference Angle Formula
Quadrant I	$0^\circ < \theta < 90^\circ$	$\theta_R = \theta$
Quadrant II	$90^\circ < \theta < 180^\circ$	$\theta_R = 180^\circ - \theta$
Quadrant III	$180^\circ < \theta < 270^\circ$	$\theta_R = \theta - 180^\circ$
Quadrant IV	$270^\circ < \theta < 360^\circ$	$\theta_R = 360^\circ - \theta$

When you use it

Use this table to find the acute reference angle for any non-quadrantal angle in standard position.

Watch out

Always subtract to or from the horizontal x-axis (180 degrees or 360 degrees), never the vertical y-axis (90 degrees or 270 degrees).

Drafted from Grade 10 Mathematics, pages 187-237, then checked twice before it went up

Sine rule, cosine rule and the area formulaFormula

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

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

$a,\,b,\,c$: the lengths of the three sides
$A,\,B,\,C$: the angles. Each one faces the side written with the same letter, so A faces a and sits between b and c

Formula	Reach for it when	Where it holds
$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$	You know two angles and any one side, or two sides and an angle opposite one of them.	Every triangle.
$a^{2} = b^{2} + c^{2} - 2bc\,\cos A$	You know two sides and the angle between them, and you want the third side.	Every triangle.
$\cos A = \dfrac{b^{2} + c^{2} - a^{2}}{2bc}$	You know all three sides and you want an angle.	Every triangle.
$\text{Area} = \tfrac{1}{2}\,ab\,\sin C$	You know two sides and the angle between them. The same triangle also gives $\tfrac{1}{2}\,bc\,\sin A$ and $\tfrac{1}{2}\,ac\,\sin B$ .	Every triangle.
$\sin A = \dfrac{\text{opposite}}{\text{hypotenuse}}$	You know the side facing the angle and the hypotenuse.	Right-angled triangles only.
$\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}$	You know the side beside the angle and the hypotenuse.	Right-angled triangles only.
$\tan A = \dfrac{\text{opposite}}{\text{adjacent}}$	You know the side facing the angle and the side beside it.	Right-angled triangles only.
$a^{2} + b^{2} = c^{2}$	You know two sides and you want the third. Here $c$ is the hypotenuse, the side facing the right angle.	Right-angled triangles only. It is the cosine rule with the angle set to 90 degrees, where the cosine is zero.

When you use it

Use these when a triangle question mixes sides and angles and the triangle has no right angle. Match what you are given against the table: two angles and any one side calls for the sine rule, while two sides with the angle between them calls for the cosine rule, and that same pair hands you the area straight away.

Watch out

Two traps, and both are about which angle belongs with which sides. The cosine rule subtracts a single product: work out $2bc\,\cos A$ as one quantity and take it away from $b^{2} + c^{2}$ . Working out $b^{2} + c^{2} - 2bc$ first and then multiplying by $\cos A$ gives a different number every time. Second, the angle has to be the one enclosed by the two sides you multiply. $\tfrac{1}{2}\,ab\,\sin C$ is the area only because $C$ sits between $a$ and $b$ , and $\cos A$ in the cosine rule is the angle between $b$ and $c$ . One more: when the sine rule gives you an angle, the calculator returns only the acute answer, so check whether 180 degrees minus that angle also fits the triangle before you accept it.

Practice questions on unit 4

10 multiple-choice questions written against this chapter, each with its answer and the reason. Try one before you open the answer.

Question 1

Convert 120° into radians.

A.2π/3
B.3π/2
C.π/3
D.4π/3

Show the answer and why

Answer: A 2π/3

Why

Use the rule that π radian = 180°. So 120° = π × 120/180 = 2π/3.

Question 2

Convert 4π/3 radian into degrees.

A.180°
B.240°
C.270°
D.300°

Show the answer and why

Answer: B 240°

Why

To change radians to degrees, multiply by 180/π. So 4π/3 × 180/π = 4 × 60 = 240°.

Question 3

What is the complementary angle of 57°?

A.33°
B.57°
C.123°
D.147°

Show the answer and why

Answer: A 33°

Why

Two angles are complementary when they add up to 90°. So the complement of 57° is 90° - 57° = 33°.

Question 4

An angle x is in standard position. Sin x is positive and cos x is negative. In which quadrant is angle x?

A.First quadrant
B.Second quadrant
C.Third quadrant
D.Fourth quadrant

Show the answer and why

Answer: B Second quadrant

Why

Sin x is positive in the first and second quadrants. Cos x is negative in the second and third quadrants. The only quadrant that fits both is the second quadrant.

Question 5

The point P(5, 12) is on the terminal side of angle θ in standard position. What is sin θ?

A.5/13
B.12/13
C.12/5
D.13/12

Show the answer and why

Answer: B 12/13

Why

First find the distance r from the origin: r = √(5² + 12²) = √169 = 13. Then sin θ = y/r = 12/13.

Question 6

Find the reference angle θR for θ = 245°.

A.25°
B.55°
C.65°
D.115°

Show the answer and why

Answer: C 65°

Why

245° is a third quadrant angle (between 180° and 270°). For a third quadrant angle, the reference angle is the angle minus 180°. So θR = 245° - 180° = 65°.

Question 7

Which of these angles is NOT co-terminal with 30°?

A.390°
B.-690°
C.750°
D.60°

Show the answer and why

Answer: D 60°

Why

Co-terminal angles differ from 30° by a multiple of 360°. 390° = 30° + 360°. -690° = 30° - 720°. 750° = 30° + 720°. But 60° - 30° = 30°, which is not a multiple of 360°, so 60° is not co-terminal with 30°.

Question 8

A tall building in Addis Ababa is 40 m high. Abebe stands on top of it and sees a car on the level ground at the foot of the building. The angle of depression to the car is 60°. How far is the car from the base of the building?

A.40√3 m
B.40/√3 m
C.20√3 m
D.40 m

Show the answer and why

Answer: B 40/√3 m

Why

The angle of depression equals the angle of elevation from the car, so the angle at the car is 60°. Using tan 60° = height/distance, we get distance = 40/tan 60° = 40/√3 m.

Question 9

Angle x is in the third quadrant and sin x = -3/5. What is cos x?

A.-4/5
B.4/5
C.-3/5
D.3/4

Show the answer and why

Answer: A -4/5

Why

Use sin²x + cos²x = 1. So cos²x = 1 - (9/25) = 16/25, which gives cos x = ±4/5. In the third quadrant cos x is negative, so cos x = -4/5.

Question 10

The minute hand of a watch is 1.5 units long. How far does its tip move in 15 minutes? Use the rule l = rθ.

A.0.75π units
B.1.5π units
C.0.5π units
D.3π units

Show the answer and why

Answer: A 0.75π units

Why

In 60 minutes the minute hand makes one full turn, so in 15 minutes it turns 15/60 = 1/4 of a turn. The angle is θ = (1/4) × 2π = π/2. Then l = r × θ = 1.5 × π/2 = 0.75π units.

The app holds 12 questions on this unit, marked as you go. These are the first 10.