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Trigonometry www.mathsrevision.com Let’s Investigate The Tangent Ratio The Tangent Angle The Sine Ratio The Sine Angle The Cosine Ratio The Cosine Angle Mixed Problems Extension www.mathsrevision.com Starter Questions 1. Find the missing value 3 ? = 4 20 2. Calculate 20% of 6000 3. What is the next three numbers in the sequence 9, 15, 21, 27, ...., ...., .... 4. Round 72 to the nearest 10 www.mathsrevision.com Trigonometry Let’s Investigate! www.mathsrevision.com Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite www.mathsrevision.com Trigonometry x° Adjacent Mathemagic! Opposite www.mathsrevision.com Trigonometry 30° Adjacent Opposite = 0.6 Adjacent Try another! Opposite www.mathsrevision.com Trigonometry 45° Adjacent Opposite = 1 Adjacent www.mathsrevision.com Trigonometry For an angle of 30°, Opposite = 0.6 Adjacent Opposite is called the tangent of an angle. Adjacent We write tan 30° = 0.6 www.mathsrevision.com Trigonometry 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 30° =0.554 0.577 Tan 29° Tan 30° 0.577 Tan 31° 0.601 Tan 32° 0.625 Tan 33° 0.649 Tan 34° 0.675 Accurate to 3 decimal places! www.mathsrevision.com Trigonometry 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! www.mathsrevision.com Trigonometry What’s the point of all this??? Don’t worry, you’re about to find out! www.mathsrevision.com Trigonometry How high is the tower? Opp 60° 12 m Opposite www.mathsrevision.com Trigonometry Copy this! 60° 12 m Adjacent www.mathsrevision.com Trigonometry Opp Tan x° = Adj Change side, change sign! Opp Tan 60° = 12 12 x Tan 60° = Opp Opp =12 x Tan 60° = 20.8m (1 d.p.) Copy this! www.mathsrevision.com Trigonometry ? 20.8m So the tower’s 20.8 m high! Don’t worry, you’ll be trying plenty of examples!! www.mathsrevision.com Starter Questions 1. Find the perimeter of the shape 2. Calculate 30% of 900 3. Find the area of the rectangle 6cm in length by 4 cm wide. 4. Name the shape. www.mathsrevision.com 3cm Opp Tan x° = Adj Opposite www.mathsrevision.com Trigonometry x° Adjacent Example www.mathsrevision.com Trigonometry Op c p Opp Tan x° = Adj 65° 8m Tan 65° = c 8 Change side, change sign! 8 x Tan 65° = c c = 8 x Tan 65° = 17.2m (1 d.p.) www.mathsrevision.com Trigonometry Now try Exercise 1. (HSDU Support Materials) Starter Questions www.mathsrevision.com 1. Name the part of the circle. 2. Calculate 60% of 300 3. If I am facing North and turn 90o clockwise, which direction am I facing 4. How many lines of symmetry has the shape. www.mathsrevision.com Using Tan to calculate angles www.mathsrevision.com Example www.mathsrevision.com Trigonometry Op p 18m x° 12m SOH CAH TOA Opp Tan x° = Adj Tan x° = 18 12 Tan x° = 1.5 ? Trigonometry www.mathsrevision.com 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 2nd Tan ⁻¹ Tan Followed by Tan Trigonometry www.mathsrevision.com Tan x° = 1.5 Press 2nd Enter 1.5 Tan ⁻¹ Tan = x = Tan ⁻¹1.5 = 56.3° (1 d.p.) www.mathsrevision.com Trigonometry Now try Exercise 2. (HSDU Support Materials) Starter Questions www.mathsrevision.com 1. 13.9 x 7 2. Calculate 23.34 x 10 3 3. of 80 4 4. Find the missing number 1, 1, 2, 3, 5, 8, ...., ...., .... www.mathsrevision.com Trigonometry Sin x° = Opposite www.mathsrevision.com The Sine Ratio x° Opp Hyp Example www.mathsrevision.com Trigonometry O Op p Opp Sin x° = Hyp Sin 34° = O 11 11cm 34° Change side, change sign! 11 x Sin 34° = O O = 11 x Sin 34° = 6.2cm (1 d.p.) www.mathsrevision.com Trigonometry Now try Exercise 3. (HSDU Support Materials) www.mathsrevision.com Starter Questions 1. 320 8 2. Calculate 20% of 360 3. Calculate 72 - 58 4. Calculate the value of the missing angle. www.mathsrevision.com 57o Using Sin to calculate angles www.mathsrevision.com Example Trigonometry www.mathsrevision.com 6m Op p 9m SOH CAH TOA x° Opp Sin x° = Hyp 6 Sin x° = 9 Sin x° = 0.667 (3 d.p.) ? Trigonometry www.mathsrevision.com 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 2nd Sin ⁻¹ Sin Followed by Sin Trigonometry www.mathsrevision.com Sin x° = 0.667 (3 d.p.) Press 2nd Enter 0.667 Sin ⁻¹ Sin = x = Sin ⁻¹0.667 = 41.8° (1 d.p.) www.mathsrevision.com Trigonometry Now try Exercise 4. (HSDU Support Materials) www.mathsrevision.com Starter Questions 1. 2.39 - 1.58 + 3.2 2. Calculate 15% of 380 3. What is the next three numbers in the sequence 2, 15, 28, 41, ...., ...., .... 4. Round 3.25 to the nearest 0.1 www.mathsrevision.com The Cosine Ratio www.mathsrevision.com Trigonometry Cos x° = x° Adjacent Adj Hyp Example Trigonometry Adj Cos x° = Hyp 40° Op p www.mathsrevision.com b b Cos 40° = 35 35mm Change side, change sign! 35 x Cos 40° = b b = 35 x Cos 40°= 26.8mm (1 d.p.) www.mathsrevision.com Trigonometry Now try Exercise 5. (HSDU Support Materials) www.mathsrevision.com Starter Questions Q1. Calculate 75% of £200 Q2. Round to 1 decimal place 2.354. Q3. How many minutes in 3hours Q4. The answer to the question is 180. What is the question. www.mathsrevision.com Using Cos to calculate angles www.mathsrevision.com Example Trigonometry Adj Cos x° = Hyp Cos x° = Op p www.mathsrevision.com SOH CAH TOA 34 45 34cm x° 45cm Cos x° = 0.756 (3 d.p.) x = Cos ⁻¹0.756 =40.9° (1 d.p.) www.mathsrevision.com Trigonometry Now try Exercise 6. (HSDU Support Materials) www.mathsrevision.com Starter Questions 1. Calculate 14 x 100 2. What kind of angle is this 3. 56.98 10 4. Name the angle that is between 180o and 360o www.mathsrevision.com The Three Ratios Sine www.mathsrevision.com Cosine Tangent Sine Sine Tangent Cosine Cosine Sine www.mathsrevision.com Trigonometry www.mathsrevision.com The Three Ratios Sin x° = Opp Hyp Cos x° = Adj Hyp Tan x° = Opp Adj www.mathsrevision.com Trigonometry Sin x° = Opp Hyp O S H O S H Cos x° = Adj Hyp A C H A C H Copy this! Tan x° = Opp Adj O T A O T A Mixed Examples Cos 20° www.mathsrevision.com Sin 36° Sin 30° Tan 27° Sin 60° Tan 40° Cos 12° Cos 79° Sin 35° www.mathsrevision.com Example 1 Trigonometry www.mathsrevision.com SOH CAH TOA Opp Sin x° = Hyp O Sin 40° = 15 O Op p 15m 40° Change side, change sign! 15 x Sin 40° = O O= 15 x Sin 40° = 9.6m (1 d.p.) Example 2 Trigonometry Adj Cos x° = Hyp b Cos 35° = 23 b 35° Op p www.mathsrevision.com SOH CAH TOA 23cm Change side, change sign! 23 x Cos 35° = b b = 23 x Cos 35° = 18.8cm (1 d.p.) Example 3 www.mathsrevision.com Trigonometry 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.) www.mathsrevision.com Trigonometry Now try Exercise 7. (HSDU Support Materials) www.mathsrevision.com Level E Starter Questions 1. Calculate 41.9 x 100 2. What kind of angle is this 3. 1.268 100 4. Name the angle that is between 0o and 90o www.mathsrevision.com www.mathsrevision.com Extension www.mathsrevision.com Example 1 www.mathsrevision.com Trigonometry 23cm Op p b SOH CAH TOA 30° Opp Sin x° = Hyp 23 Sin 30° = b ? www.mathsrevision.com Trigonometry 23 Sin 30° = b Change sides, change signs! 23 b= Sin 30° (This means b = 23 ÷ Sin 30º) b= 46 cm Example 2 Trigonometry 7m 50° Adj Cos x° = Hyp p 7 Cos 50° = Change sides, change signs! p 7 p= Cos 50° Op p www.mathsrevision.com SOH CAH TOA p= 10.9m (1 d.p.) Example 3 www.mathsrevision.com Trigonometry 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.)