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Presents Let’s Investigate The Tangent ratio The Sine ratio The Cosine ratio The three ratios Extension Let’s Investigate! Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite x° Adjacent Mathemagic! Opposite 30° Adjacent Opposite = 0.6 Adjacent Try another! Opposite 45° Adjacent Opposite = 1 Adjacent For an angle of 30°, Opposite Adjacent Opposite = 0.6 Adjacent is called the tangent of an angle. We write tan 30° = 0.6 The ancient Greeks discovered this and repeated this for all 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! Now-a-days we can use calculators instead of tables to find the Tan of an angle. On your calculator press Followed by 30, and press Tan = 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! How high is the tower? h 60° 12 m Copy this! Opposite h 60° 12 m Adjacent Opp Tan x° = Adj h Tan 60° = 12 Copy this! Change side, change sign! 12 x Tan 60° = h h = 12 x Tan 60° = 20.8m (1 d.p.) So the tower’s 20.8 m high! ? 20.8m Don’t worry, you’ll be trying plenty of examples!! The Tangent Ratio Opp Tan x° = Adj Opposite x° Adjacent Example Op c p 65° 8m Opp Tan x° = Adj Tan 65° = c 8 Change side, change sign! 8 x Tan 65° = c c = 8 x Tan 65° = 17.2m (1 d.p.) Now try Exercise 1. (HSDU Support Materials) Using Tan to calculate angles Example Op p 18m x° 12m SOH CAH TOA Opp Tan x° = Adj Tan x° = 18 12 Tan x° = 1.5 ? Tan x° = 1.5 How do we find x°? We need to use Tan ⁻¹on the calculator. Tan ⁻¹is written above To get this press Tan ⁻¹ Tan 2nd Followed by Tan Tan x° = 1.5 Press 2nd Tan ⁻¹ Tan Enter 1.5 = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) Now try Exercise 2. (HSDU Support Materials) The Sine Ratio Sin x° = Opp Hyp Opposite x° h Op p Example 11cm 34° Opp Sin x° = Hyp h Sin 34° = 11 Change side, change sign! 11 x Sin 34° = h h = 11 x Sin 34° = 6.2cm (1 d.p.) Now try Exercise 3. (HSDU Support Materials) Using Sin to calculate angles 6m Op p Example 9m SOH CAH TOA x° Opp Sin x° = Hyp 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.) ? Sin x° =0.667 (3 d.p.) How do we find x°? We need to use Sin ⁻¹on the calculator. Sin ⁻¹is written above To get this press Sin ⁻¹ Sin 2nd Followed by Sin Sin x° = 0.667 (3 d.p.) Press 2nd Sin ⁻¹ Sin Enter 0.667 = x = Sin ⁻¹0.667 = 41.8° (1 d.p.) Now try Exercise 4. (HSDU Support Materials) The Cosine Ratio Cos x° = Adj Hyp x° Adjacent b 40° Example Op 35mm Adj Cos x° = Hyp b Cos 40° = 35 Change side, change sign! 35 x Cos 40° = b b = 35 x Cos 40°= 26.8mm (1 d.p.) Now try Exercise 5. (HSDU Support Materials) Using Cos to calculate angles 34cm x° Example Op SOH CAH TOA 45cm Adj Cos x° = Hyp 34 Cos x° = 45 Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =40.9° (1 d.p.) Now try Exercise 6. (HSDU Support Materials) Tangent Sine Cosine The Three Ratios Sine Sine Tangent Cosine Cosine Sine The Ratios Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj The Ratios Sin x° = Opp Hyp Cos x° = Adj Hyp Copy this! Tan x° = Opp Adj O S H A C H O T A SOH CAH TOA Tan 27° Sin 36° Cos 20° Mixed Examples Sin 30° Sin 60° Tan 40° Cos 12° Cos 79° Sin 35° h Op p Example 1 15m SOH CAH TOA 40° Opp Sin x° = Hyp h Sin 40° = 15 Change side, change sign! 15 x Sin 40° = h h = 15 x Sin 40° = 9.6m (1 d.p.) b 35° Example 2 Op SOH CAH TOA 23cm Adj Cos x° = Hyp b Cos 35° = 23 Change side, change sign! 23 x Cos 35° = b b = 23 x Cos 35° = 18.8cm (1 d.p.) Example 3 Op c p 60° 15m SOH CAH TOA Opp Tan x° = Adj c Tan 60° = 15 Change side, change sign! 15 x Tan 60° = c c = 15 x Tan 60° = 26.0m (1 d.p.) Now try Exercise 7. (HSDU Support Materials) Extension 23cm Op p Example 1 b SOH CAH TOA 30° Opp Sin x° = Hyp 23 Sin 30° = b ? 23 Sin 30° = b Change sides, change signs! 23 b= Sin 30° (This means b = 23 ÷ Sin 30º) b= 46 cm 7m 50° Example 2 Op SOH CAH TOA p Adj Cos x° = Hyp 7 Cos 50° = Change sides, change signs! p 7 p= Cos 50° p= 10.9m (1 d.p.) Example 3 Op 9m p 55° d SOH CAH TOA Opp Tan x° = Adj 9 Tan 55° = d 9 d= Tan 55° Change sides, change signs! d= 6.3m (1 d.p.) © K Hughes 2001