Unit notes and practice questions

Grade 9 Mathematics unit 5: Introduction to Trigonometry

Everything here is for unit 5 of the Grade 9 Mathematics textbook: what the chapter covers, the short notes, practice questions with their answers, and the lesson video where one exists.

4
Cards: 10
Practice questions on unit 5: 162–180
The textbook pages for unit 5

What unit 5 covers

1.Revision on Right-angled Triangles
2.The Trigonometric Ratios

Short notes for unit 5

The cards filed under this chapter on the Grade 9 Mathematics page. Open one at a time.

Confusing Opposite and Adjacent SidesCommon mistake

When you use it

Check this before setting up sine, cosine, or tangent ratios for a given acute angle.

Watch out

The hypotenuse is always opposite the 90 degree angle, but opposite and adjacent sides swap places when you switch from one acute angle to the other.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Pythagoras TheoremFormula

b^2 + p^2 = h^2

b^2 + p^2 = h^2

$h$: hypotenuse $m$
$b$: base $m$
$p$: perpendicular side $m$

When you use it

Use this to find an unknown side length of a right-angled triangle or to verify if a triangle is a right-angled triangle.

Watch out

The hypotenuse h must always be the longest side opposite the 90 degree angle. Never substitute a leg in place of h.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Trigonometric RatiosFormula

\sin A = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan A = \frac{\text{opposite}}{\text{adjacent}}

\sin A = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos A = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan A = \frac{\text{opposite}}{\text{adjacent}}

$A$: acute angle $^\circ$
$\text{opposite}$: side opposite to angle A $m$
$\text{adjacent}$: side adjacent to angle A $m$
$\text{hypotenuse}$: hypotenuse $m$

When you use it

Use these ratios to relate an acute angle with the side lengths in a right-angled triangle.

Watch out

The opposite and adjacent sides depend on which acute angle is chosen. Make sure they are identified with respect to angle A.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Trigonometric Values of Special AnglesReference tableA

Ratio	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$
$\sin A$	$0$	$\frac{1}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos A$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{2}}$	$\frac{1}{2}$	$0$
$\tan A$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

When you use it

Use these exact values when evaluating trigonometric expressions with standard acute angles without a table.

Watch out

The value of tan 90 degrees is undefined because division by zero occurs.

Drafted from Grade 9 Mathematics, pages 162-180, then checked twice before it went up

Practice questions on unit 5

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

Question 1

In a right-angled triangle, what is the special name given to the side opposite to the 90 degree angle?

A.Base
B.Perpendicular
C.Hypotenuse
D.Adjacent

Show the answer and why

Answer: C Hypotenuse

Why

By definition, the side opposite to the right angle (90 degrees) in a right-angled triangle is called the hypotenuse, and it is always the longest side.

Question 2

What is the maximum degree measure of an interior angle in a right-angled triangle?

A.45 degrees
B.60 degrees
C.90 degrees
D.180 degrees

Show the answer and why

Answer: C 90 degrees

Why

A right-angled triangle has one angle equal to 90 degrees, and the sum of the other two acute angles is 90 degrees. Therefore, the maximum interior angle measure is 90 degrees.

Question 3

The word trigonometry is derived from two Greek words: trigonon and metron. What does trigonon mean?

A.Measure
B.Triangle
C.Angle
D.Circle

Show the answer and why

Answer: B Triangle

Why

According to the textbook, trigonon means triangle and metron means measure.

Question 4

For an acute angle A in a right-angled triangle, how is the tangent ratio (tan A) defined?

A.Opposite side divided by adjacent side
B.Adjacent side divided by hypotenuse
C.Opposite side divided by hypotenuse
D.Hypotenuse divided by adjacent side

Show the answer and why

Answer: A Opposite side divided by adjacent side

Why

By definition, tan A = (side opposite to angle A) / (side adjacent to angle A).

Question 5

Which of the following sets of side lengths can form a right-angled triangle?

A.5, 6, 2
B.4, 6, 8
C.1, 2, √3
D.2, 3, 4

Show the answer and why

Answer: C 1, 2, √3

Why

Using the converse of Pythagoras theorem: for side lengths 1, √3, and 2 (where 2 is the longest side), 1² + (√3)² = 1 + 3 = 4, and 2² = 4. Since LHS = RHS, these sides form a right-angled triangle.

Question 6

An equilateral triangle has sides of length 6 units. What is the height of this triangle?

A.3√3 units
B.3√2 units
C.2√5 units
D.3 units

Show the answer and why

Answer: A 3√3 units

Why

Drawing a perpendicular height splits the base of 6 into two segments of 3 units each. By Pythagoras theorem, h² + 3² = 6², which gives h² + 9 = 36, so h² = 27, and h = 3√3 units.

Question 7

What is the exact value of tan 60°?

A.1 / √3
B.1
C.√3
D.√3 / 2

Show the answer and why

Answer: C √3

Why

In a standard 30°-60°-90° right-angled triangle, the side opposite to 60° is √3 and the adjacent side is 1. Thus, tan 60° = √3 / 1 = √3.

Question 8

As the acute angle A increases from 0° to 90°, what happens to the values of sin A and cos A?

A.sin A increases from 0 to 1, while cos A decreases from 1 to 0
B.sin A decreases from 1 to 0, while cos A increases from 0 to 1
C.Both sin A and cos A increase from 0 to 1
D.Both sin A and cos A decrease from 1 to 0

Show the answer and why

Answer: A sin A increases from 0 to 1, while cos A decreases from 1 to 0

Why

As noted in the textbook remark, as ∠A increases from 0° to 90°, sin A increases from 0 to 1 while cos A decreases from 1 to 0.

Question 9

In a right-angled triangle, if an acute angle A has sin A = 3/5, what is the value of tan A?

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

Show the answer and why

Answer: B 3/4

Why

Since sin A = opposite / hypotenuse = 3/5, let opposite = 3 and hypotenuse = 5. By Pythagoras theorem, adjacent = √(5² - 3²) = √(25 - 9) = √16 = 4. Therefore, tan A = opposite / adjacent = 3/4.

Question 10

If A is an acute angle in a right-angled triangle and tan A = 1 / √2, what is the value of cos A?

A.1 / √3
B.√2 / √3
C.√3 / √2
D.1 / 2

Show the answer and why

Answer: B √2 / √3

Why

With tan A = opposite / adjacent = 1 / √2, opposite = 1 and adjacent = √2. Hypotenuse = √(1² + (√2)²) = √(1 + 2) = √3. Therefore, cos A = adjacent / hypotenuse = √2 / √3.

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