Trigonometric Ratios

1. Trigonometric Ratios In Trigonometry, the comparison is between sides of a triangle. Used to find a side of a right triangle given 1 side and 1 acute angle

2. Θ this is the symbol for an unknown angle measure. It’s name is ‘Theta’. Easy way to remember trig ratios: SOH CAH TOA Three Trigonometric Ratios • Sine – abbreviated ‘sin’. • Ratio: sin θ = opposite side • hypotenuse • Cosine - abbreviated ‘cos’. • Ratio: cos θ = adjacent side • hypotenuse • Tangent - abbreviated ‘tan’. • Ratio: tan θ = opposite side • adjacent side

4. Let’s practice… Write the ratio for sin A Sin A = o= a h c Write the ratio for cos A Cos A = a = b h c Write the ratio for tan A Tan A = o = a a b B c a C b A Let’s switch angles: Find the sin, cos and tan for Angle B: Tan B = b a Sin B = b c Cos B = a c

6. Make sure you have a calculator… Set your calculator to ‘Degree’….. MODE (next to 2nd button) Degree (third line down… highlight it) 2nd Quit

7. Using trig ratios in equations Remember back in 1st grade when you had to solve: 12 = x What did you do? 6 (6) (6) 72 = x Remember back in 3rd grade when x was in the denominator? 12 = 6 What did you do? x (x) (x) 12x = 6 __ __ 12 12 x = 1/2

8. Trigonometric Ratios • When do we use them? • On right triangles that are NOT 45-45-90 or 30-60-90 Find: tan 45 1 Why? tan = opp hyp

9. Example: Find the value of x. Step 1: Mark the “Angle of Perspective”. Step 2: Label the sides (Hyp / Opp / Adj). Step 3: Select a trigonometry ratio (sin/ cos / tan). Sin = Step 4: Substitute the values into the equation. Sin 25 = If the variable is on top multiply If the variable is on the bottom Divide Step 5: Solve the equation : Hyp opp Angle of Perspective Adj x = 12Sin (25 x = 5.07 cm

10. 34° 15 cm x cm Opposite and hypotenuse Ask yourself: In relation to the angle, what pieces do I have? Ask yourself: What trig ratio uses Opposite and Hypotenuse? SINE Set up the equation and solve: (15) (15) Sin 34 = x 15 (15)Sin 34 = x 8.39 cm = x

11. 53° 12 cm Opposite and adjacent Ask yourself: In relation to the angle, what pieces do I have? Ask yourself: What trig ratio uses Opposite and adjacent? x cm tangent Set up the equation and solve: (12) (12) Tan 53 = x 12 (12)tan 53 = x 15.92 cm = x

12. x cm Adjacent and hypotenuse Ask yourself: In relation to the angle, what pieces do I have? 68° Ask yourself: What trig ratio uses adjacent and hypotnuse? 18 cm cosine Set up the equation and solve: (x) (x) Cos 68 = 18 x (x)Cos 68 = 18 _____ _____ cos 68 cos 68 X = 18 cos 68 X = 48.05 cm

13. Let’s practice… Find an angle that has a tangent (ratio) of 2 3 Round your answer to the nearest degree. C 2cm B 3cm A Process: I want to find an ANGLE I was given the sides (ratio) Tangent is opp adj TAN-1(2/3) = 34°

14. 8 A 4 Practice some more… Find tan A: 24.19 12 A 21 Tan A = opp/adj = 12/21 Tan A = .5714 Find tan A: 8 Tan A = 8/4 = 2

15. 42 cm 22 cm Opposite and hypotenuse THIS IS IMPORTANT!! Ask yourself: What trig ratio uses opposite and hypotenuse? θ This time, you’re looking for theta. Ask yourself: In relation to the angle, what pieces do I have? sine Set up the equation (remember you’re looking for theta): Sin θ = 22 42 Remember to use the inverse function when you find theta Sin -122 = θ 42 31.59°= θ

16. θ THIS IS IMPORTANT!! Ask yourself: What trig ratio uses the parts I was given? 22 cm You’re still looking for theta. 17 cm tangent Set it up, solve it, tell me what you get. tan θ = 17 22 tan -117 = θ 22 37.69°= θ

17. Using trig ratios in equations Remember back in 1st grade when you had to solve: 12 = x What did you do? 6 (6) (6) 72 = x Remember back in 3rd grade when x was in the denominator? 12 = 6 What did you do? x (x) (x) 12x = 6 __ __ 12 12 x = 1/2