Let’s Investigate The Tangent Ratio The Tangent Angle The Sine Ratio The Sine Angle The Cosine Ratio The Cosine Angle Mixed Problems Extension

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Trigonometry Let’s Investigate! www.mathsrevision.com

Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite hypotenuse x° Adjacent

Mathemagic! Opposite hypotenuse 30° Adjacent Opposite = 0.6 Adjacent

Try another! Opposite hypotenuse 45° Adjacent Opposite = 1 Adjacent

Opposite Adjacent = 0.6 For an angle of 30°, Opposite Adjacent is called the tangent of an angle. We write tan 30° = 0.6

The ancient Greeks discovered this and repeated this for possible angles. Tan 25° 0.466 Tan 26° 0.488 Tan 27° 0.510 Tan 28° 0.532 Tan 29° 0.554 Tan 30° 0.577 Tan 31° 0.601 Tan 32° 0.625 Tan 33° 0.649 Tan 34° 0.675 Tan 30° = 0.577 Accurate to 3 decimal places!

On your calculator press Now-a-days we can use calculators instead of tables to find the Tan of an angle. On your calculator press Tan Followed by 30, and press = Notice that your calculator is incredibly accurate!! Accurate to 9 decimal places!

What’s the point of all this??? Don’t worry, you’re about to find out!

Opp How high is the tower? 60° 12 m

Copy this! Opposite hypotenuse 60° 12 m Adjacent

Opp Tan x° = Adj Opp Tan 60° = 12 12 x Tan 60° = Opp Opp = Copy this! Opp Tan x° = Change side, change sign! Adj Opp Tan 60° = 12 12 x Tan 60° = Opp Opp = 12 x Tan 60° = 20.8m (1 d.p.)

? 20.8m So the tower’s 20.8 m high! Don’t worry, you’ll be trying plenty of examples!!

Starter Questions 3cm www.mathsrevision.com www.mathsrevision.com

Opp Tan x° = Adj Opposite x° Adjacent

Opp Tan x° = Adj c Tan 65° = 8 8 x Tan 65° = c c = 8 x Tan 65° Example Opp Hyp Opp c Tan x° = Adj 65° c Tan 65° = Change side, change sign! 8m 8 Adj 8 x Tan 65° = c c = 8 x Tan 65° = 17.2m (1 d.p.)

(HSDU Support Materials) Now try Exercise 1. (HSDU Support Materials)

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Using Tan to calculate angles www.mathsrevision.com

? SOH CAH TOA Opp Tan x° = Adj 18 Tan x° = 12 Tan x° = 1.5 Example Opp Hyp 18m Opp ? Tan x° = Adj x° 12m 18 Tan x° = Adj 12 Tan x° = 1.5

We need to use Tan ⁻¹on the calculator. = 1.5 Tan x° How do we find x°? Tan Tan ⁻¹ Tan ⁻¹is written above To get this press 2nd Followed by Tan

= 1.5 Tan x° Press Enter 1.5 = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) Tan ⁻¹

(HSDU Support Materials) Now try Exercise 2. (HSDU Support Materials)

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The Sine Ratio Opp Sin x° = Hyp Opposite hypotenuse x°

Opp Sin x° = Hyp O Sin 34° = 11 = O 11 x Sin 34° O = 11 x Sin 34° Example Hyp 11cm O Opp Opp Sin x° = 34° Hyp O Sin 34° = Change side, change sign! 11 = O 11 x Sin 34° O = 11 x Sin 34° = 6.2cm (1 d.p.)

(HSDU Support Materials) Now try Exercise 3. (HSDU Support Materials)

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Using Sin to calculate angles www.mathsrevision.com

? SOH CAH TOA Opp Sin x° = Hyp 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.) Example Hyp 9m 6m Opp SOH CAH TOA x° Opp ? Sin x° = Hyp 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.)

We need to use Sin ⁻¹on the calculator. How do we find x°? =0.667 (3 d.p.) Sin x° We need to use Sin ⁻¹on the calculator. How do we find x°? Sin Sin ⁻¹ Sin ⁻¹is written above To get this press 2nd Followed by Sin

= 0.667 (3 d.p.) Sin x° Press Enter 0.667 = x = Sin ⁻¹0.667 2nd Enter 0.667 = x = Sin ⁻¹0.667 = 41.8° (1 d.p.)

(HSDU Support Materials) Now try Exercise 4. (HSDU Support Materials)

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The Cosine Ratio Adj Cos x° = Hyp hypotenuse x° Adjacent

Adj Cos x° = Hyp b Cos 40° = 35 35 x Cos 40° = b b = 35 x Cos 40° Example b Adj 40° Adj Cos x° = Opp Hyp Hyp 35mm b Cos 40° = Change side, change sign! 35 35 x Cos 40° = b b = 35 x Cos 40° = 26.8mm (1 d.p.)

(HSDU Support Materials) Now try Exercise 5. (HSDU Support Materials)

Starter Questions www.mathsrevision.com Q1. Calculate Q2. Round to 1 decimal place 2.354. Q3. How many minutes in 3hours www.mathsrevision.com Q4. The answer to the question is 180. What is the question. www.mathsrevision.com

Using Cos to calculate angles www.mathsrevision.com

SOH CAH TOA Adj Cos x° = Hyp 34 Cos x° = 45 Cos x° = 0.756 (3 d.p.) Example SOH CAH TOA Adj 34cm x° Adj Cos x° = Opp Hyp Hyp 45cm 34 Cos x° = 45 Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =40.9° (1 d.p.)

(HSDU Support Materials) Now try Exercise 6. (HSDU Support Materials)

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The Three Ratios www.mathsrevision.com Sine Tangent Cosine Sine Sine

The Three Ratios Sin x° = Opp Hyp Cos x° = Adj Tan x° =

O S H A C H O T A CAH TOA SOH Sin x° = Opp Hyp Cos x° = Adj Tan x° = Copy this! Sin x° = Opp Hyp Cos x° = Adj Tan x° = O S H A C H O T A CAH TOA SOH

Mixed Examples www.mathsrevision.com Cos 20° Tan 27° Sin 36° Sin 60°

SOH CAH TOA Opp Sin x° = Hyp O Sin 40° = 15 15 x Sin 40° = O O = Example 1 SOH CAH TOA 15m Hyp O Opp Opp Sin x° = Hyp 40° O Sin 40° = Change side, change sign! 15 15 x Sin 40° = O O = 15 x Sin 40° = 9.6m (1 d.p.)

SOH CAH TOA Adj Cos x° = Hyp b Cos 35° = 23 23 x Cos 35° = b b = Example 2 SOH CAH TOA b Adj 35° Adj Cos x° = Opp Hyp Hyp 23cm b Cos 35° = Change side, change sign! 23 23 x Cos 35° = b b = 23 x Cos 35° = 18.8cm (1 d.p.)

SOH CAH TOA Opp Tan x° = Adj c Tan 60° = 15 15 x Tan 60° = c c = Example 3 SOH CAH TOA Opp Hyp c Opp Tan x° = Adj 60° c 15m Tan 60° = Change side, change sign! 15 Adj 15 x Tan 60° = c c = 15 x Tan 60° = 26.0m (1 d.p.)

(HSDU Support Materials) Now try Exercise 7. (HSDU Support Materials)

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? SOH CAH TOA Opp Sin x° = Hyp 23 Sin 30° = b Example 1 Hyp b 23cm Opp

23 Sin 30° = b 23 b= Sin 30° (This means b = 23 ÷ Sin 30º) b= 46 cm Change sides, change signs! b 23 b= Sin 30° (This means b = 23 ÷ Sin 30º) b= 46 cm

SOH CAH TOA Adj Cos x° = Hyp 7 Cos 50° = p 7 p= Cos 50° p= Example 2 SOH CAH TOA 7m Adj 50° Adj Opp Cos x° = Hyp Hyp p 7 Cos 50° = Change sides, change signs! p 7 p= Cos 50° p= 10.9m (1 d.p.)

SOH CAH TOA Opp Tan x° = Adj 9 Tan 55° = d 9 d= Tan 55° d= Example 3 SOH CAH TOA Opp Hyp Opp 9m Tan x° = Adj 55° 9 Tan 55° = Change sides, change signs! d Adj d 9 d= Tan 55° d= 6.3m (1 d.p.)