LemonmathSelf-study sheet

Grade 9 · Lesson 4.1

Trigonometric ratios of acute angles

Name: Date: Time spent: min

Goals

Write the sine, cosine, tangent and cotangent of an acute angle in a right-angled triangle.
Learn the trigonometric ratios of $30^\circ$ , $45^\circ$ and $60^\circ$ , and use the link between complementary angles.
Use a calculator to find trigonometric ratios, and to find an angle from a ratio.

Remember

$\sin\alpha=\dfrac{\text{opp}}{\text{hyp}}$ , $\cos\alpha=\dfrac{\text{adj}}{\text{hyp}}$ , $\tan\alpha=\dfrac{\text{opp}}{\text{adj}}$ , $\cot\alpha=\dfrac{\text{adj}}{\text{opp}}$ .
The ratios depend only on the angle; $0<\sin\alpha<1$ , $0<\cos\alpha<1$ ; $\tan\alpha\cdot\cot\alpha=1$ .
$\sin30^\circ=\cos60^\circ=\dfrac12$ , $\sin45^\circ=\cos45^\circ=\dfrac{\sqrt2}2$ , $\sin60^\circ=\cos30^\circ=\dfrac{\sqrt3}2$ , $\tan45^\circ=1$ , $\tan60^\circ=\sqrt3$ , $\tan30^\circ=\dfrac{\sqrt3}3$ .
Complementary angles: the sine of one is the cosine of the other, and the tangent of one is the cotangent of the other.
Calculator: degree mode; SHIFT + sin/cos/tan to find an angle.

Exercises

Exercise 1

Triangle $ABC$ has a right angle at $A$ , $AB=3$ and $AC=4$ . Find the trigonometric ratios of angle $B$ .

Exercise 2

Triangle $ABC$ has a right angle at $A$ , $BC=17$ and $AB=8$ . Find the trigonometric ratios of angle $C$ .

Exercise 3

Evaluate: a) $\sin45^\circ\cdot\cos45^\circ$ ; b) $\tan60^\circ-\cot30^\circ+\tan45^\circ$ .

Exercise 4

Write as trigonometric ratios of angles smaller than $45^\circ$ : $\sin70^\circ$ , $\cos58^\circ$ , $\tan80^\circ$ , $\cot47^\circ$ .

Exercise 5

Without a calculator, evaluate $M=\dfrac{\sin32^\circ}{\cos58^\circ}+\tan17^\circ\cdot\tan73^\circ$ .

Exercise 6

Use a calculator:

a) Find $\sin54^\circ$ and $\tan20^\circ$ (rounded to four decimal places).

b) Find the acute angle $\alpha$ (to the nearest degree) given that $\tan\alpha=2$ ; given that $\sin\alpha=0.25$ .

Exercise 7

A " $10\%$ gradient" road sign means that for every $100$ m horizontally, the road rises $10$ m. Building guidance says a wheelchair ramp should be no steeper than $1:12$ (a rise of $1$ m over $12$ m horizontally). Find the angle each slope makes with the horizontal (to the nearest degree).

Exercise 8

Triangle $ABC$ has a right angle at $A$ , $BC=10$ cm and $\sin B=0.6$ .

a) Find $AC$ .

b) Find $AB$ .

c) Find $\tan B$ and $\cos C$ .

d) Find the size of angle $B$ (to the nearest degree).

Exercise 9

Triangle $ABC$ has a right angle at $A$ , $AB=1$ and $AC=2$ . Find $\sin B$ and $\cos B$ , with denominators free of roots.

Exercise 10 Challenge

Triangle $ABC$ is isosceles with $AB=AC$ , $BC=12$ cm and $\angle B=30^\circ$ . Draw the altitude $AH$ . Find $AH$ and $AB$ (in a neat exact form).

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Triangle

ABC

has a right angle at

A

. Then

\sin B

equals

\dfrac{AC}{BC}

.B.

\dfrac{AB}{BC}

.C.

\dfrac{AC}{AB}

.D.

\dfrac{AB}{AC}

Question 2.

\tan60^\circ

equals

\sqrt3

.B.

\dfrac{\sqrt3}3

.C.

\dfrac{\sqrt3}2

.D.

\dfrac12

Question 3.

\cos52^\circ

equals

\sin38^\circ

.B.

\cos38^\circ

.C.

\sin52^\circ

.D.

\tan38^\circ

Question 4. Triangle

ABC

has a right angle at

A

AB=12

and

BC=13

. Then

\cos B

equals

\dfrac{12}{13}

.B.

\dfrac{5}{13}

.C.

\dfrac{5}{12}

.D.

\dfrac{13}{12}

Part II. True or false

Question 1. Triangle

ABC

has a right angle at

A

AB=\sqrt3

and

AC=1

. Decide whether each statement is true or false.

a) $BC=2$ .	T / F
b) $\sin B=\dfrac12$ .	T / F
c) $\tan B=\sqrt3$ .	T / F
d) $\angle C=60^\circ$ .	T / F

Part III. Short answer

Question 1. Evaluate

2\sin30^\circ+3\tan45^\circ-\cot45^\circ

Question 2. A road has an "

8\%

gradient" sign. Find the angle the road makes with the horizontal, in degrees (give your answer to 1 decimal place).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: the definitions of the four ratios; Pythagoras' theorem.

Steps:

Find $BC$ .
For angle $B$ : the opposite side is $AC$ , the adjacent side is $AB$ .

Exercise 2.

What you need: $AC^2=BC^2-AB^2$ ; for angle $C$ , the opposite side is $AB$ .

Steps:

Find $AC$ .
Write the four ratios.

Exercise 3.

What you need: the table of special-angle ratios.

Steps:

Substitute the values.
Work it out.

Exercise 4.

What you need: if $\alpha+\beta=90^\circ$ , then $\sin\alpha=\cos\beta$ and $\tan\alpha=\cot\beta$ .

Steps:

Find the complement of each angle.
Swap sin ↔ cos and tan ↔ cot.

Exercise 5.

What you need: complementary angles and $\tan\alpha\cdot\cot\alpha=1$ .

Steps:

Write $\cos58^\circ=\sin32^\circ$ .
Write $\tan73^\circ=\cot17^\circ$ .

Exercise 6.

What you need: degree mode; SHIFT + tan (or sin) to find an angle.

Steps:

Check that the calculator is in mode D.
Key in and round.

Exercise 7.

What you need: the angle of slope $\alpha$ has $\tan\alpha=\dfrac{\text{rise}}{\text{horizontal distance}}$ .

Steps:

Write $\tan\alpha$ for each slope.
Press SHIFT tan.

Exercise 8.

What you need: $\sin B=\dfrac{AC}{BC}$ ; Pythagoras' theorem; $\cos C=\sin B$ because the angles are complementary.

Steps:

$AC=BC\cdot\sin B$ .
$AB=\sqrt{BC^2-AC^2}$ .
Write the ratios; press SHIFT sin $0.6$ .

Exercise 9.

What you need: rationalising the denominator (see transforming and simplifying expressions with square roots).

Steps:

Find $BC=\sqrt5$ .
Write the ratios, then rationalise the denominators.

Exercise 10.

What you need: in an isosceles triangle the altitude is also a median, so $BH=6$ cm; $\tan30^\circ=\dfrac{\sqrt3}3$ , $\cos30^\circ=\dfrac{\sqrt3}2$ .

Steps:

In the right-angled triangle $ABH$ : $\tan B=\dfrac{AH}{BH}$ .
$\cos B=\dfrac{BH}{AB}$ gives $AB$ ; rationalise the denominator.

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