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!!!!!!!