Trigonometry www mathsrevision com Lets Investigate The Tangent

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

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Trigonometry Let’s Investigate! www. mathsrevision. com

Trigonometry means “triangle” and “measurement”. We will be using right-angled triangles. Opposite www. mathsrevision. com Trigonometry hy po te nu se x° Adjacent

Mathemagic! Opposite www. mathsrevision. com Trigonometry hyp o ten use 30° Adjacent Opposite = 0. 6 Adjacent

Try another! Opposite www. mathsrevision. com Trigonometry hyp o ten use 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 29° 0. 554 Tan 30° = 0. 577 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 hy po te nu se 60° 12 m Adjacent Copy this!

www. mathsrevision. com Trigonometry Opp Tan x° = Adj Change side, change sign! Opp Tan 60° = 12 12 x Tan 60° = Opp =12 x Tan 60° = 20. 8 m (1 d. p. ) Copy this!

www. mathsrevision. com Trigonometry ? 20. 8 m So the tower’s 20. 8 m high! Don’t worry, you’ll be trying plenty of examples!!

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Opp Tan x° = Adj Opposite www. mathsrevision. com Trigonometry x° Adjacent

Example www. mathsrevision. com Trigonometry Hyp 65° 8 m Adj Opp c Opp Tan x° = Adj Tan 65° = c 8 Change side, change sign! 8 x Tan 65° = c c = 8 x Tan 65° = 17. 2 m (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 1. (HSDU Support Materials)

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Example www. mathsrevision. com Trigonometry S OH C A H T OA Hyp Opp 18 m x° 12 m Adj 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 2 nd Tan ⁻¹ Tan Followed by Tan

Trigonometry www. mathsrevision. com Tan x° = 1. 5 Press 2 nd Enter 1. 5 Tan ⁻¹ Tan = x = Tan ⁻¹ 1. 5 = 56. 3° (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 2. (HSDU Support Materials)

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Trigonometry Sin x° = Opposite www. mathsrevision. com The Sine Ratio hy po te nu se x° Opp Hyp

Example www. mathsrevision. com Trigonometry O Opp Sin x° = Hyp Sin 34° = 11 cm Hyp 34° O 11 Change side, change sign! 11 x Sin 34° = O O = 11 x Sin 34° = 6. 2 cm (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 3. (HSDU Support Materials)

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Using Sin to calculate angles www. mathsrevision. com

Example Trigonometry www. mathsrevision. com 6 m Opp 9 m Hyp 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 2 nd Sin ⁻¹ Sin Followed by Sin

Trigonometry www. mathsrevision. com Sin x° = 0. 667 (3 d. p. ) Press 2 nd Sin ⁻¹ Sin Enter 0. 667 x = Sin ⁻¹ 0. 667 = = 41. 8° (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 4. (HSDU Support Materials)

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The Cosine Ratio www. mathsrevision. com Trigonometry Cos x° = hy po te nu se x° Adjacent Adj Hyp

Example Adj Cos x° = Hyp b Adj 40° Opp www. mathsrevision. com Trigonometry b Cos 40° = 35 Hyp 35 mm Change side, change sign! 35 x Cos 40° = b b = 35 x Cos 40°= 26. 8 mm (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 5. (HSDU Support Materials)

www. mathsrevision. com Starter Questions Q 1. Calculate Q 2. Round to 1 decimal place 2. 354. Q 3. How many minutes in 3 hours Q 4. 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° = Opp www. mathsrevision. com S OH C A H T OA 34 45 34 cm Adj x° Hyp 45 cm Cos x° = 0. 756 (3 d. p. ) x = Cos ⁻¹ 0. 756 =40. 9° (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 6. (HSDU Support Materials)

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The Three Ratios www. mathsrevision. com Sine Cosine Tangent Sine Tangent 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 Cos x° = Adj Hyp A C H Copy this! Tan x° = Opp Adj 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 S OH C A H T OA Opp Sin x° = Hyp O Sin 40° = 15 O Opp 15 m Hyp 40° Change side, change sign! 15 x Sin 40° = O O= 15 x Sin 40° = 9. 6 m (1 d. p. )

Example 2 Trigonometry Adj Cos x° = Hyp b Cos 35° = 23 b Adj 35° Opp www. mathsrevision. com S OH C A H T OA Hyp 23 cm Change side, change sign! 23 x Cos 35° = b b = 23 x Cos 35° = 18. 8 cm (1 d. p. )

Example 3 www. mathsrevision. com Trigonometry Hyp 60° 15 m Adj Opp c S OH C A H T OA Opp Tan x° = Adj c Tan 60° = 15 Change side, change sign! 15 x Tan 60° = c c = 15 x Tan 60° = 26. 0 m (1 d. p. )

www. mathsrevision. com Trigonometry Now try Exercise 7. (HSDU Support Materials)

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Example 1 www. mathsrevision. com Trigonometry 23 cm Opp b Hyp S OH C A H T OA 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 Adj Cos x° = Hyp 7 m Adj 50° Opp www. mathsrevision. com S OH C A H T OA p Hyp 7 Cos 50° = Change sides, change signs! p 7 p= Cos 50° p= 10. 9 m (1 d. p. )

Example 3 Trigonometry www. mathsrevision. com S OH C A H T OA Opp 9 m Hyp Opp Tan x° = Adj 55° Adj d 9 Tan 55° = d 9 d= Tan 55° Change sides, change signs! d= 6. 3 m (1 d. p. )