Trigonometric Ratios A RATIO is a comparison of

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 Name “say” Abbreviation Abbrev. Ratio of an angle measure Sine Cosine tangent Sin Cos Tan Sinθ = opposite side cosθ = adjacent side hypotenuse tanθ =opposite side adjacent side

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

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

Make sure you have a calculator… I want to find Use these calculator keys sin, cos or tan ratio Angle measure Set your calculator to ‘Degree’…. . MODE (next to 2 nd button) Degree (third line down… highlight it) 2 nd Quit SIN COS TAN SIN-1 COS-1 TAN-1

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

Practice some more… Find tan A: 24. 19 12 Tan A = opp/adj = 12/21 Tan A =. 5714 A 21 Find tan A: Tan A = 8/4 = 2 8 4 A

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

Ask yourself: In relation to the angle, what pieces do I have? 34° 15 cm Opposite and hypotenuse Ask yourself: x cm What trig ratio uses Opposite and Hypotenuse? SINE Set up the equation and solve: (15) Sin 34 = x 15 (15)Sin 34 = x 8. 39 cm = x

Ask yourself: In relation to the angle, what pieces do I have? 53° 12 cm Opposite and adjacent x cm Ask yourself: What trig ratio uses Opposite and adjacent? tangent Set up the equation and solve: (12) Tan 53 = x 12 (12)tan 53 = x 15. 92 cm = x

x cm Ask yourself: In relation to the angle, what pieces do I have? Adjacent and hypotenuse 68° 18 cm Ask yourself: What trig ratio uses adjacent and hypotnuse? cosine Set up the equation and solve: (x) (x) Cos 68 = 18 x (x)Cos 68 = 18 _____ cos 68 X = 18 X = 48. 05 cm cos 68

42 cm 22 cm θ This time, you’re looking for theta. TH IS Ask yourself: IS In relation to the angle, what pieces IMPO RT do I have? Opposite and hypotenuse AN T! Ask yourself: ! What trig ratio uses opposite and hypotenuse? 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 -1 22 = θ 42 31. 59°= θ

You’re still looking for theta. TH IS θ IS IMP Ask yourself: 22 cm 17 cm What trig ratio uses the parts I was given? tangent Set it up, solve it, tell me what you get. tan θ = 17 22 tan -1 17 = θ 22 37. 69°= θ OR TA NT !!

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

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 2 nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle.

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 c 2 = 32 + 22 c 2 = 9 + 4 c 2 = 13 c = √ 13 c ≈ 3. 6 Pythagorean Theorem Substitute values Simplify Find the positive square root Use a calculator to approximate

Example 1 continued • Then use a calculator to find the measure of B: 2 nd 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 about 33. 7° and 56. 3°.

Ex. 2: Solving a Right Triangle (h) • Solve the right triangle. 25° You are looking for Round decimals to the opposite and nearest tenth. hypotenuse which is the sin ratio. sin H = 13 sin 25° = opp. hyp. h 13 13(0. 4226) ≈ h 5. 5 ≈ h Set up the correct ratio 13 Substitute values/multiply by reciprocal Substitute value from table or calculator Use your calculator to approximate.

Ex. 2: Solving a Right Triangle (g) • Solve the right triangle. 25° Round decimals to the You are looking for nearest tenth. adjacent and hypotenuse which is the cosine ratio. cos G = adj. hyp. g 13 cos 25° = 13 13(0. 9063) ≈ g 11. 8 ≈ h Set up the correct ratio 13 Substitute values/multiply by reciprocal Substitute value from table or calculator Use your calculator to approximate.

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: • 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. Glide = x° 15. 7 miles 59 miles tan x° = opp. adj. 15. 7 Use correct ratio Substitute values 59 Key in calculator 2 nd 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 • 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. The shuttle’s altitude is about 1. 7 miles. 5 miles tan 19° = 5 tan 19° = opp. adj. Use correct ratio h Substitute values 5 h 5 5 Isolate h by multiplying by 5. 1. 7 ≈ h Approximate using calculator