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Trigonometry

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**Trigonometry is a method of finding out an unknown angle or side in a right angled triangle**

Both the triangles below are similar because: The angles are the same but the sides are different

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**Trigonometry is a method of finding out an unknown angle or side in a right angled triangle**

Both the triangles below are similar because: The angles are the same but the sides are different

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**Trigonometry is a method of finding out an unknown angle or side in a right angled triangle**

Both the triangles below are similar because: The angles are the same but the sides are different

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**If we measure the height and the base:**

Small triangle Large triangle For both triangles 5 cm 8 cm 10 cm 16 cm

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This angle is in fact 320 So as long as the value of then this angle will always be 320

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**This is the idea behind trigonometry**

If we know 2 sides then we can find the angles in the triangle How do we know the angle is 320 ? We can use our calculator which has been programmed to work out the angle.

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**We don`t have to know the height and the base it can be any 2 sides**

Depending on which 2 sides are known then we use a different button on the calculator Names are given to the 3 sides which all refer to the angle we are trying to find

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**Opposite, Adjacent and Hypotenuse**

The names are: Opposite, Adjacent and Hypotenuse Opposite means on the other side from the angle we need. Adjacent means next to the angle we need. Hypotenuse means the side opposite the right angle Hypotenuse Opposite X Adjacent

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**Identify the names of the sides of these right angled-triangles given angle k**

opposite b opposite a b hypotenuse c c adjacent a hypotenuse k opposite a k adjacent c adjacent b c opposite k hypotenuse hypotenuse b k a adjacent

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**Opposite, Adjacent and Hypotenuse **

In each case label all the sides of the triangles as Opposite (O), Adjacent (A) and Hypotenuse (H) with relation to the angle marked as “X”. X X X x X X X X X X X

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**Using the Opposite (O), Adjacent (A) and **

Hypotenuse (H) to work out the missing angle The calculator has 3 buttons which are used to find the missing angle: Sin – short for Sine Cos – short for Cosine Tan – short for Tangent

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**Deciding which button to use depends on which sides are given**

SOH CAH TOA Memory Aid Some Old Horses Sin Opposite Hypotenuse Can Always Hear Cos Adjacent Hypotenuse Their Owners Approaching Tan Opposite Adjacent Or invent one of your own

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**SOH CAH TOA S O H C A H T O A Divide it up into three groups**

Place each group of three in a triangle starting in the bottom left of each triangle S O H C A H T O A

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Trigonometric Ratios S O H SOH C A H CAH T O A TOA

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**SOH CAH TOA Cos (x) = x Example 1 What have we got and need to find?**

We need an angle – x. We have the Hypotenuse and Adjacent side. 10 cm 25 cm x Looking at the phrase, we can use C A H Hypotenuse Cos (x) = Adjacent Hypotenuse Adjacent

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Cos (x) = Adjacent Hypotenuse 10 cm 25cm x Replace A and H by 10 and 25 Hypotenuse Cos (x) = = 0.4 We now need to convert this to an angle in degrees using the Cos-1 button!!! Adjacent x = Cos –1(0.4) = 66.42o We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.

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**SOH CAH TOA Tan (x) = x Example 2 What have we got and need to find?**

We need an angle – x. We have the Opposite and Adjacent side. Looking at the phrase, we can use TOA 15 cm Tan (x) = Opposite Adjacent Opposite x 20 cm Adjacent

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**We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons.**

Tan (x) = Opposite Adjacent Replace O and A by 15 and 20 Tan (x) = = 0.75 We now need to convert this to an angle in degrees using the Tan-1 button!!! 15 cm Opposite x x = Tan –1(0.75) = 36.67o 20 cm Adjacent

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**SOH CAH TOA Sin (x) = x Example 3 What have we got and need to find?**

We need an angle – x. We have the Hypotenuse and Opposite side. Looking at the phrase, we can use S O H Hypotenuse 12 cm 8 cm Sin (x) = Opposite Hypotenuse Opposite x

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Sin (x) = Opposite Hypotenuse We always find the angle using either the Cos–1, Sin–1 or Tan–1 buttons. Replace O and H by 8 and 12 Sin (x) = = 0.666 Hypotenuse We now need to convert this to an angle in degrees using the Sin-1 button!!! 12 cm 8 cm Opposite x x = Sin –1(0.666) = 41.81o

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**Using Trigonometry to Find a Missing Side**

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Trigonometric Ratios S O H SOH C A H CAH T O A TOA

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**Trigonometric Ratios S O H**

The triangle can also be used to find either the opposite side or the hypotenuse

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**Trigonometric Ratios C A H**

The triangle can also be used to find either the adjacent side or the hypotenuse

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**Trigonometric Ratios O A T**

TOA The triangle can also be used to find either the adjacent side or the opposite

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**H S O H Example 1 SOH CAH TOA What have we got and need to find?**

We need the Hypotenuse H We have an angle and the Opposite O Hypotenuse Looking at the phrase we can use S O H S O H Opposite Hypotenuse = Opposite Sin (angle)

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**H Opposite Hypotenuse = Sin (angle) Replace O by 3 and (angle) by 60o**

Sin (60o) 60o Use the Sin button on your calculator to find this value H H = Hypotenuse H = ….. H = 3.46 m to 2 d.p. 3 m Opposite

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**A T O A Example 2 SOH CAH TOA What have we got and need to find?**

We need the Adjacent A 40o We have an angle and the Opposite O A Adjacent Looking at the phrase we can use T O A T O A 3 m Opposite

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**A Replace O by 3 and (angle) by 40o 3 40o Tan (40o) H = Adjacent H =**

Use the Tan button on your calculator to find this value A H = Adjacent H = 3.575….. 3 m H = 3.58 m to 2 d.p. Opposite

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**8 C A H Example 3 SOH CAH TOA What have we got and need to find?**

We need the Adjacent A We have an angle and the Hypotenuse H 8 Hypotenuse Looking at the phrase we can use C A H 70o A C A H Adjacent

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**8 Replace H by 8 and (angle) by 70o A = cos70 x 8 H = 0.342 x 8**

Use the Cos button on your calculator to find this value 8 H = x 8 Hypotenuse H = 2.736….. 70o A H = 2.74 m to 2 d.p. Adjacent

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