Trigonometric Ratios

Trigonometric Ratios A RATIO is a comparison of two numbers. For example; boys to girls cats : dogs right : wrong. In Trigonometry, the comparison is between sides of a triangle ( right triangle).

Trig. Ratios

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 Some Old Horse Caught Another Horse Taking Oats Away

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 = Sin B = Cos B =

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

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°

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

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

Using trig ratios in equations : 12 = x 6

34° 15 cm x cm Opposite and hypotenuse 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 Ask yourself: In relation to the angle, what pieces do I have? 8.39 cm = x

53° 12 cm Opposite and adjacent 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 Ask yourself: In relation to the angle, what pieces do I have? 15.92 cm = x

x cm Adjacent and hypotenuse 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 Ask yourself: In relation to the angle, what pieces do I have?

42 cm 22 cm Opposite and hypotenuse THIS IS IMPORTANT!! Ask yourself: What trig ratio uses opposite and hypotenuse? θ 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 This time, you’re looking for theta. Ask yourself: In relation to the angle, what pieces do I have? 31.59°= θ

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

Types of Angles • The angle that your line of sight makes with a line drawn horizontally. • Angle of Elevation • Angle of Depression

SOA CAH TOA SOH CAH TOA

Solving a right triangle • Every right triangle has one right angle, two acute angles, one hypotenuse and two legs. To solve a right triangle, means to determine the measures of all six (6) parts. You can solve a right triangle if the following one of the two situations exist: • Two side lengths • One side length and one acute angle measure

WRITE THIS DOWN!!! The expression sin-1 x is read as “the inverse sine of x.” • On your calculator, this means you will be punching the 2nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle. • In general, for an acute angle A: • If sin A = x, then sin-1 x = mA • If cos A = y, then cos-1 y = mA • If tan A = z, then tan-1 z = mA

Example 1: • Solve the right triangle. Round the decimals to the nearest tenth. HINT: Start by using the Pythagorean Theorem. You have side a and side b. You don’t have the hypotenuse which is side c—directly across from the right angle.

Example 1: (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem Substitute values c2 = 32 + 22 Simplify c2 = 9 + 4 Simplify c2 = 13 Find the positive square root c = √13 Use a calculator to approximate c ≈3.6

Example 1 continued • Then use a calculator to find the measure of B: 2nd function Tangent button 2 Divided by symbol 3 ≈ 33.7°

Finally • Because A and B are complements, you can write mA = 90° - mB ≈ 90° - 33.7° = 56.3° The side lengths of the triangle are 2, 3 and √13, or about 3.6. The triangle has one right angle and two acute angles whose measure are about 33.7° and 56.3°.

Ex. 2: Solving a Right Triangle (h) 25° • Solve the right triangle. Round decimals to the nearest tenth. Set up the correct ratio Substitute values/multiply by reciprocal Substitute value from table or calculator Use your calculator to approximate.

Ex. 2: Solving a Right Triangle (g) 25°

Using Right Triangles in Real Life • Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes. • A. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.

Solution: Glide  = x° 15.7 miles 59 miles • You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle. opp. tan x° = Use correct ratio adj. 15.7 Substitute values tan x° = 59 Key in calculator 2nd function, tan 15.7/59 ≈ 14.9  When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9°.

B. Solution Glide  = 19° h 5 miles • When the space shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth. opp. tan 19° = Use correct ratio adj. h Substitute values tan 19° = 5 h 5 Isolate h by multiplying by 5. 5 tan 19° = 5  The shuttle’s altitude is about 1.7 miles. 1.7 ≈ h Approximate using calculator

Trigonometric Ratios