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

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Let’s Investigate! Trigonometry

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

Trigonometry 30° Adjacent Opposite hypotenuse Opposite Adjacent = 0.6 Mathemagic!

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

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

Trigonometry Tan 25° Tan 26° Tan 27° Tan 28° Tan 29° Tan 30° Tan 31° Tan 32° Tan 33° Tan 34° Tan 30° = Accurate to 3 decimal places! The ancient Greeks discovered this and repeated this for possible angles.

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

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

Trigonometry 12 m How high is the tower? Opp 60°

Trigonometry 60° 12 m Adjacent Opposite hypotenuse Copy this!

Trigonometry Tan x° = Opp Adj Tan 60° = Opp 12 = Opp12 x Tan 60° Opp =12 x Tan 60°= 20.8m (1 d.p.) Change side, change sign! Copy this!

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

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Trigonometry Adj x°x°x°x° Tan x° = O p p o s i t e Opp Adjacent

Trigonometry Example 65° Tan x° = Op p Adj Hyp c 8m Tan 65° = c 8 = c8 x Tan 65° c =8 x Tan 65°= 17.2m (1 d.p.) Adj Change side, change sign!

Trigonometry Now try Exercise 1. ( HSDU Support Materials)

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Using Tan to calculate angles

Trigonometry Example x°x°x°x° Tan x° = Op p Adj Hyp S O H C A H T O A 12m Tan x° = = 1.5Tan x° Adj 18m ?

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

Trigonometry x = Tan ⁻ ¹ 1.5 = 56.3° (1 d.p.) = 1.5Tan x° 2 nd Tan Tan ⁻ ¹ Press Enter = 1.5

Trigonometry Now try Exercise 2. ( HSDU Support Materials)

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Trigonometry The Sine Ratio x°x°x°x° Sin x° = O p p o s i t e Opp Hyp h y p o t e n u s e

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

Trigonometry Now try Exercise 3. ( HSDU Support Materials)

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Using Sin to calculate angles

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

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

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

Trigonometry Now try Exercise 4. ( HSDU Support Materials)

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Trigonometry The Cosine Ratio Cos x° = Adjacent Adj x°x°x°x° Hyp h y p o t e n u s e

Trigonometry Example 40° Cos x° = Op p Adj Hyp b 35mm Cos 40° = b 35 = b35 x Cos 40° b =35 x Cos 40°= 26.8mm (1 d.p.) Adj Change side, change sign!

Trigonometry Now try Exercise 5. ( HSDU Support Materials)

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

Using Cos to calculate angles

Trigonometry Example x°x°x°x° Cos x° = Op p Adj Hyp S O H C A H T O A 45cm Cos x° = = (3 d.p.)Cos x° x = Cos ⁻ ¹0.756 =40.9° (1 d.p.) Adj 34cm

Trigonometry Now try Exercise 6. ( HSDU Support Materials)

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The Three Ratios Cosine Sine Tangent Sine Tangent Cosine Sine

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

Trigonometry Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj CAHCAHTOATOASOHSOH A C H O T A O S H Copy this!

Mixed Examples Cos 12° Sin 60° Tan 27° Sin 30° Sin 35° Tan 40° Cos 20° Cos 79° Sin 36°

Trigonometry Example 1 40° Sin x° = Op p Hyp S O H C A H T O A O 15m Sin 40° = O 15 = O15 x Sin 40° O =15 x Sin 40°= 9.6m (1 d.p.) Change side, change sign!

Trigonometry Example 2 35° Cos x° = Op p Adj Hyp S O H C A H T O A b 23cm Cos 35° = b 23 = b23 x Cos 35° b =23 x Cos 35°= 18.8cm (1 d.p.) Adj Change side, change sign!

Trigonometry Example 3 60° Tan x° = Op p Adj Hyp S O H C A H T O A c 15m Tan 60° = c 15 = c15 x Tan 60° c =15 x Tan 60°= 26.0m (1 d.p.) Adj Change side, change sign!

Trigonometry Now try Exercise 7. ( HSDU Support Materials)

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Extension

Trigonometry Example 1 30° Sin x° = Op p Hyp S O H C A H T O A 23cm b Sin 30° = 23 b ?

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

Trigonometry Example 2 50° Cos x° = Op p Adj Hyp S O H C A H T O A 7m p Cos 50° = 7 p p= 10.9m (1 d.p.) Adj Change sides, change signs! Cos 50° 7

Trigonometry Example 3 55° Tan x° = Op p Adj Hyp S O H C A H T O A 9m d Adj Tan 55° = 9 d d= 6.3m (1 d.p.) Change sides, change signs! Tan 55° 9