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).

CCSS: G.SRT.7 EXPLAIN and USE the relationship between the sine and cosine of complementary angles.

Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2.Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6.Attend to precision. • 7. Look for and make use of structure. • 8.Look for and express regularity in repeated reasoning.

Warm up • Solve the equations: • A) 0.875 = x/18 • B) 24/y = .5 • C) y/25 = .96

E.Q: How can we find the sin, cosine, and the tangent of an acute angle? How do we use trigonometric ratios to solve real-life problems?

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

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

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 • 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

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

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

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

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

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

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

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

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

Indirect Measurement

SOA CAH TOA SOA 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

E.Q • How do we use right triangles to solve real life problems?

Note: • As you learned in Lesson 9.5, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. As you will see in this lesson, once you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle.

WRITE THIS DOWN!!! • 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 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.

Solve the right triangle. Round the decimals to the nearest tenth. Example 1: 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°.

Solve the right triangle. Round decimals to the nearest tenth. opp. sin H = hyp. h 13 13 sin 25° = 13 Ex. 2: Solving a Right Triangle (h) 25° You are looking for opposite and hypotenuse which is the sin ratio. Set up the correct ratio Substitute values/multiply by reciprocal Substitute value from table or calculator 13(0.4226) ≈ h Use your calculator to approximate. 5.5 ≈ h

Solve the right triangle. Round decimals to the nearest tenth. adj. cos G = hyp. g 13 13 cos 25° = 13 Ex. 2: Solving a Right Triangle (g) 25° You are looking for adjacent and hypotenuse which is the cosine ratio. Set up the correct ratio Substitute values/multiply by reciprocal Substitute value from table or calculator 13(0.9063) ≈ g 11.8 ≈ h Use your calculator to approximate.

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. Using Right Triangles in Real Life

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. Solution: Glide  = x° 15.7 miles 59 miles 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°.

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. B. Solution Glide  = 19° h 5 miles 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