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# Right Triangle Trigonometry

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SAM, section 4.3

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### Right Triangle Trigonometry

1. 1. Chapter 4: Trigonometric Functions Section 4.3 (pg. 277-287)
2. 2. • If we want to convert degrees to radians we multiply by… π 180 • If we want to convert radians to degrees we multiply by… 180 π A degree is an angle measure, a radian is an arc measure
3. 3. What is the Pythagorean Theorem? • In a right triangle, the length of one leg squared plus the length of the second leg squared equals the squared length of the hypotenuse. So what does that mean in English? a2 + b2 = c2
4. 4. - A right triangle is a triangle with a right angle. Let’s look at one of the acute angles of the triangle, which we will call theta = θ θ - The side opposite the right angle is called the hypotenuse - The side opposite θ is called the opposite side (makes sense right?) - The side next to θ is called the adjacent sideAdjacent Opposite Using the lengths of these 3 sides, you can form six ratios that define the six trigonometric functions of the acute angle θ
5. 5. sin(x) cos(x) tan(x)
6. 6. cosecant(x) secant(x) cotangent(x)
7. 7. - If θ is the acute angle of a right triangle, then we can find the values of sin(θ), cos(θ) and tan(θ) using the side lengths of the triangle like so: sin(θ) = cos(θ) = tan(θ) = opposite opposite hypotenuse hypotenuse adjacent adjacent
8. 8. θ Find: sin(θ) = = cos(θ) = = tan(θ) = = opposite hypotenuse opposite adjacent adjacent hypotenuse 4 5 3 5 4 3
9. 9. I N E P P O S I T E P P O S I T E Y P O T E N U S E Y P O T E N U S E O S I N E A N G E N T D J A C E N T D J A C E N T
10. 10. - If θ is the acute angle of a right triangle, then we can also find the values of cosecant(θ), secant(θ) and cotangent(θ) using the side lengths of the triangle like so: cosecant(θ) = secant(θ) = cotangent(θ) = hypotenuse adjacent opposite adjacent opposite hypotenuse
11. 11. θ Find: csc(θ) = = sec(θ) = = cot(θ) = = hypotenuse opposite adjacent opposite hypotenuse adjacent 5 4 5 3 3 4
12. 12. Take a few minutes and make some conjectures with your table group about the relationships between the 6 trig functions Write down at least one conjecture
13. 13. - In the earlier example, you were given the side lengths of a triangle and asked to find sin(θ), cos(θ), and tan(θ). Sometimes you will be asked to find sin, cos, tan for a given acute angle, like θ = 45⁰ - How can you find the value of sin(45⁰)? • We could use a calculator: sin(45⁰) = .7071067812 • But that’s just an approximation, what if we wanted the exact value, not the decimal approximation? We can construct a right triangle with 45⁰ as one of its acute angles
14. 14. Find the exact value of sin(45⁰) • We will start by drawing a triangle with 45⁰ as one of its acute angles. The length of the adjacent side will be 1 unit. 1 45⁰ • Now we can use a little bit of Geometry to help us out. Since two angles of the triangle are 90⁰ and 45⁰, the other angle has to be … 45⁰ 45⁰ • Since the base angles are congruent, the legs must be congruent also, so the other side length is … 1 1 • Finally, the Pythagorean Theorem tells us that the hypotenuse is … √2 √2
15. 15. Find the exact value of sin(45⁰) 1 45⁰ 45⁰ 1 √2 • Now we are almost done! We can use SOH CAH TOA to find sin(45⁰) sin(45⁰) = = opp hyp 1 √2 Now find cos(45⁰) and tan(45⁰) and check your answer with a calculator Use your calculator to make sure that this is the answer we got earlier
16. 16. 1 45⁰ 45⁰ 1 √2 This triangle has a special name … It is called a 45-45-90 right triangle, and we use it to find sin(45⁰), cos(45⁰) and tan(45⁰)
17. 17. 1 60⁰ 30⁰ 2 This triangle also has a special name It is called a 30-60-90 right triangle, and we can use it to find trig values involving 30⁰ and 60⁰ √3 Find: tan(60⁰) cos(30⁰) sin(30⁰) cos(60⁰) Now check your answers with a calculator
18. 18. • Pioneer’s Clock Tower is 40 ft high, and you are standing 40 ft away staring at the top of the tower. At what angle is your head looking upward? o Let’s draw a picture. We know you are 40 ft away and the tower is 40 ft high. We will call the angle your head is looking up θ. θ 40 40 o We have the opposite and the adjacent sides of the right triangle, so we can say: tan(θ) = 40 40 = 1 And we know from our 45-45-90 triangle that θ must be … 45⁰
19. 19. Brendan Gibbons kicks the football at an angle of 60⁰ when he is 45 yards away from the field goal post. How far did the football travel through the air? 60⁰ 45 yards o Since this is a right triangle, we can set up a trig equation. Should we use sin(60⁰), cos(60⁰), or tan(60⁰)? cos(60⁰) = adj hyp 45 x = o And now since cos(60⁰) = ½ we can say… 1 45 2 x = o And we solve for x …