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Warm – Up: 2/4 Convert from radians to degrees.

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Presentation on theme: "Warm – Up: 2/4 Convert from radians to degrees. "— Presentation transcript:

1 Warm – Up: 2/4 Convert from radians to degrees. 𝜋 5 7𝜋 2
Convert from degrees to radians. 27° 134°

2 Homework: What questions do you have? What problems are you unsure of?
What don’t you understand?

3 Trigonometric Functions of Acute Angles
Section 4.2 – Day 1

4 Trigonometric Functions
Let 𝜽 be an acute angle in the right triangle ∆𝑨𝑩𝑪. Then, Reference Triangle 𝑠𝑖𝑛𝑒 𝜃 = sin 𝜃 = 𝑜𝑝𝑝 ℎ𝑦𝑝 𝑐𝑜𝑠𝑖𝑛𝑒 𝜃 = cos 𝜃 = 𝑎𝑑𝑗 ℎ𝑦𝑝 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = tan 𝜃 = 𝑜𝑝𝑝 𝑎𝑑𝑗 𝑐𝑜𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = csc 𝜃 = ℎ𝑦𝑝 𝑜𝑝𝑝 𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = s𝑒𝑐 𝜃 = ℎ𝑦𝑝 𝑎𝑑𝑗 𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = cot 𝜃 = 𝑎𝑑𝑗 𝑜𝑝𝑝

5 Pythagorean Theorem In a right triangle with sides 𝑎 and 𝑏 and hypotenuse 𝑐, 𝑎 2 + 𝑏 2 = 𝑐 2

6 45 – 45 – 90 Triangle To find the other angle: 180°−90°−45°=45°
To find the other side: = 𝑐 2 2= 𝑐 =𝑐 Now find the trig functions: sin 45° = 𝑜𝑝𝑝 ℎ𝑦𝑝 = ∙ = cos 45° = 𝑎𝑑𝑗 ℎ𝑦𝑝 = ∙ = tan 45° = 𝑜𝑝𝑝 𝑎𝑑𝑗 = 1 1 =1 csc 45° = ℎ𝑦𝑝 𝑜𝑝𝑝 = = 2 sec 45° = ℎ𝑦𝑝 𝑎𝑑𝑗 = = 2 cot 45° = 𝑎𝑑𝑗 𝑜𝑝𝑝 = 1 1 =1 45 – 45 – 90 Triangle 45° 2 1 45° 1

7 30 – 60 – 90 Triangle To find the other angle: 180°−90°−30°=60°
To find the other side: 1 2 + 𝑏 2 = 𝑏 2 =4 𝑏 2 =3→𝑏= 3 Now find the trig functions: sin 30° = 𝑜𝑝𝑝 ℎ𝑦𝑝 = 1 2 cos 30° = 𝑎𝑑𝑗 ℎ𝑦𝑝 = tan 30° = 𝑜𝑝𝑝 𝑎𝑑𝑗 = ∙ = csc 30° = ℎ𝑦𝑝 𝑜𝑝𝑝 =2 sec 30° = ℎ𝑦𝑝 𝑎𝑑𝑗 = ∙ = cot 30° = 𝑎𝑑𝑗 𝑜𝑝𝑝 = 3 30 – 60 – 90 Triangle 30° 2 3 60° 1

8 Exercises #1−8 Directions: Evaluate all six trigonometric functions of 𝜃: sin 𝜃 = cos 𝜃 = 5 13 tan 𝜃 = 12 5 csc 𝜃 = sec 𝜃 = 13 5 cot 𝜃 = 5 12 𝜃 13 5 12

9 Exercise #1−8 Directions: Evaluate all six trigonometric functions of 𝜃: sin 𝜃 = 6 8 = 3 4 cos 𝜃 = tan 𝜃 = = 3 7 ∙ 7 7 = csc 𝜃 = 8 6 = 4 3 sec 𝜃 = = 4 7 ∙ 7 7 = cot 𝜃 = = 7 3 8 6 𝜃 2 7 First, find the missing side: 6 2 + 𝑏 2 = 𝑏 2 =64 𝑏 2 =28 𝑏= 28 = 4 ∙ 7 =2 7

10 Homework – Due: 2/5 P. 352 – 353: #1, 5 and 8

11 Warm – Up: 2/5 Find secant, cosecant and tangent of the triangle below. 𝜃 17 8 15

12 Homework: What questions do you have? What problems are you unsure of?
What don’t you understand?

13 Trigonometric Functions of Acute Angles
Section 4.2 – Day 2

14 Trigonometric Functions
Let 𝜽 be an acute angle in the right triangle ∆𝑨𝑩𝑪. Then, Reference Triangle 𝑠𝑖𝑛𝑒 𝜃 = sin 𝜃 = 𝑜𝑝𝑝 ℎ𝑦𝑝 𝑐𝑜𝑠𝑖𝑛𝑒 𝜃 = cos 𝜃 = 𝑎𝑑𝑗 ℎ𝑦𝑝 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = tan 𝜃 = 𝑜𝑝𝑝 𝑎𝑑𝑗 𝑐𝑜𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = csc 𝜃 = ℎ𝑦𝑝 𝑜𝑝𝑝 𝑠𝑒𝑐𝑎𝑛𝑡 𝜃 = s𝑒𝑐 𝜃 = ℎ𝑦𝑝 𝑎𝑑𝑗 𝑐𝑜𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝜃 = cot 𝜃 = 𝑎𝑑𝑗 𝑜𝑝𝑝

15 Exercises 9-18: Assume that 𝜃 is an acute angle in a right triangle satisfying the given conditions. Evaluate the remaining trigonometric functions. cos 𝜃 = 𝟐 𝟏𝟎 𝟕 tan 𝜃 = = ∙ → = ∙10 = 𝟑 𝟏𝟎 𝟐𝟎 csc 𝜃 = 𝟕 𝟑 sec 𝜃 = = ∙ → = ∙10 = 𝟕 𝟏𝟎 𝟐𝟎 cot 𝜃 = 𝟐 𝟏𝟎 𝟑 sin 𝜃 = 3 7 𝜃 7 3 𝑏 2 10 First, find the missing side: 3 2 + 𝑏 2 = 𝑏 2 =49 𝑏 2 =40 𝑏= 40 = 4 ∙ 10 =2 10

16 Exercises 9-18: Assume that 𝜃 is an acute angle in a right triangle satisfying the given conditions. Evaluate the remaining trigonometric functions. sin 𝜃 = 𝟗 𝟐𝟑 cos 𝜃 = 𝟖 𝟕 𝟐𝟑 tan 𝜃 = = ∙ → = ∙7 = 𝟗 𝟕 𝟓𝟔 sec 𝜃 = = ∙ → = ∙7 = 𝟐𝟑 𝟕 𝟓𝟔 cot 𝜃 = 𝟖 𝟕 𝟗 csc 𝜃 = 23 9 𝜃 23 9 𝑏 8 7 First, find the missing side: 9 2 + 𝑏 2 = 𝑏 2 =529 𝑏 2 =448 𝑏= 448 = 64 ∙ 7 =8 7

17 Exercises 25 – 35: Directions: Evaluate using a calculator. Be sure the calculator is in the correct mode. Give answers correct to three decimal places. sin 𝜋 15 cos 8° tan 23°42′ csc 19° sec 1.24 cot 𝜋 8 0.208 0.990 0.439 3.072 1.000 2.414

18 Exercises 46 – 54: Directions: Solve for the variable shown. 𝑥 15 𝑦 32
34° 𝑥 15 57° 𝑦 32 Hypotenuse Opposite Adjacent Opposite Step one: Label your sides. Step two: Pick trig function. sin 34° = 15 𝑥 𝑥 sin 34 °=15 𝑥= 15 sin 34° ≈26.82 Step one: Label your sides. Step two: Pick trig function. tan 57° = 32 𝑦 𝑦 tan 57 °=32 𝑥= 32 tan 57° ≈20.78

19 P. 352 – 353: #16 and 18 #25 – 33 (odd) and #36 #46 – #54 (even)
Homework – Due: 2/5 P. 352 – 353: #16 and 18 #25 – 33 (odd) and #36 #46 – #54 (even) #57!!!!!!!


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