Transcript 4.1 – Right Triangle Trigonometry

4.1 – Right Triangle Trigonometry
Sin Ɵ = opp
hyp
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Sin Ɵ = opp
hyp
Cos Ɵ = adj
hyp
Csc Ɵ = hyp
opp
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Cos Ɵ = adj
hyp
Sec Ɵ = hyp
adj
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Cos Ɵ = adj
hyp
Sec Ɵ = hyp
adj
Tan Ɵ opp
adj
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Cos Ɵ = adj
hyp
Sec Ɵ = hyp
adj
Tan Ɵ opp
adj
Cot Ɵ = adj
opp
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Cos Ɵ = adj
hyp
Sec Ɵ = hyp
adj
Tan Ɵ opp
adj
Cot Ɵ = adj
opp
Csc = 1
sin
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Cos Ɵ = adj
hyp
Sec Ɵ = hyp
adj
Tan Ɵ opp
adj
Cot Ɵ = adj
opp
Csc = 1
sin
Sec = 1
cos
Sin Ɵ = opp
hyp
Csc Ɵ = hyp
opp
Cos Ɵ = adj
hyp
Sec Ɵ = hyp
adj
Tan Ɵ opp
adj
Cot Ɵ = adj
opp
Csc = 1
sin
Sec = 1
cos
Cot = 1
tan
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
8
17
Ɵ
15
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
8
17
Ɵ
15
Sin Ɵ = opp
hyp
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Csc Ɵ = hyp
opp
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Csc Ɵ = hyp = 17
opp 8
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj
hyp
Csc Ɵ = hyp = 17
opp 8
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Csc Ɵ = hyp = 17
opp 8
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Csc Ɵ = hyp = 17
opp 8
Sec Ɵ = hyp
adj
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Csc Ɵ = hyp = 17
opp 8
Sec Ɵ = hyp = 17
adj 15
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
17
8
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Tan Ɵ = opp
adj
Csc Ɵ = hyp = 17
opp 8
Sec Ɵ = hyp = 17
adj 15
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
8
17
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Tan Ɵ = opp = 8
adj 15
Csc Ɵ = hyp = 17
opp 8
Sec Ɵ = hyp = 17
adj 15
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
8
17
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Tan Ɵ = opp = 8
adj 15
Csc Ɵ = hyp = 17
opp 8
Sec Ɵ = hyp = 17
adj 15
Cot Ɵ = adj
opp
Ex. 1 Find the exact values of the six trigonometric
functions of Ɵ.
8
17
Ɵ
15
Sin Ɵ = opp = 8
hyp 17
Cos Ɵ = adj = 15
hyp 17
Tan Ɵ = opp = 8
adj 15
Csc Ɵ = hyp = 17
opp 8
Sec Ɵ = hyp = 17
adj 15
Cot Ɵ = adj = 15
opp 8
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
Ɵ
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
Ɵ
2
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
Ɵ
2
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp
hyp
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp
adj
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp
opp
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_
opp √21
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_ = 5√21
opp √21 21
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_ = 5√21
opp √21 21
Sec Ɵ = hyp
adj
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_ = 5√21
opp √21 21
Sec Ɵ = hyp = 5
adj 2
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_ = 5√21
opp √21 21
Sec Ɵ = hyp = 5
adj 2
Cot Ɵ = adj
opp
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_ = 5√21
opp √21 21
Sec Ɵ = hyp = 5
adj 2
Cot Ɵ = adj = 2_
opp √21
Ex. 2 If Cos Ɵ = 2/5, find the exact values of the five
remaining trigonometric functions for Ɵ.
5
√21
Ɵ
2
Sin Ɵ = opp = √21
hyp
5
Tan Ɵ = opp = √21
adj
2
Csc Ɵ = hyp = 5_ = 5√21
opp √21 21
Sec Ɵ = hyp = 5
adj 2
Cot Ɵ = adj = 2_ = 2√21
opp √21 21
Ex. 3 Find the value of x. Round to the nearest tenth, if
necessary.
18
x
42
Ex. 3 Find the value of x. Round to the nearest tenth, if
necessary.
18
x
Cos Ɵ = adj
hyp
42
Ex. 3 Find the value of x. Round to the nearest tenth, if
necessary.
18
x
Cos Ɵ = adj
hyp
Cos 42° = x
18
42
Ex. 3 Find the value of x. Round to the nearest tenth, if
necessary.
18
x
Cos Ɵ = adj
hyp
Cos 42° = x
18
18Cos 42° = x
42
Ex. 3 Find the value of x. Round to the nearest tenth, if
necessary.
18
x
Cos Ɵ = adj
hyp
Cos 42° = x
18
18Cos 42° = x
13.4 ≈ x
42
Ex. 4 Find the measure of angle Ɵ. Round to the nearest
degree, if necessary.
26
Ɵ
11
Ex. 4 Find the measure of angle Ɵ. Round to the nearest
degree, if necessary.
26
Ɵ
11
Tan Ɵ = opp
adj
Ex. 4 Find the measure of angle Ɵ. Round to the nearest
degree, if necessary.
26
Ɵ
11
Tan Ɵ = opp
adj
Tan Ɵ = 26
11
Ex. 4 Find the measure of angle Ɵ. Round to the nearest
degree, if necessary.
26
Ɵ
11
Tan Ɵ = opp
adj
Tan Ɵ = 26
11
Ɵ = tan-1(26/11)
Ex. 4 Find the measure of angle Ɵ. Round to the nearest
degree, if necessary.
26
Ɵ
11
Tan Ɵ = opp
adj
Tan Ɵ = 26
11
Ɵ = tan-1(26/11)
Ɵ ≈ 67°
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
Cos Ɵ = adj
hyp
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
Cos Ɵ = adj
hyp
Cos 35° = 10
z
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
Cos Ɵ = adj
hyp
Cos 35° = 10
z
z = 10___
Cos 35°
10
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
10
Cos Ɵ = adj
hyp
Cos 35° = 10
z
z = 10___ ≈ 12.2
Cos 35°
X
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
10
X
Cos Ɵ = adj
35° + Y = 90 °
hyp
Cos 35° = 10
z
z = 10___ ≈ 12.2
Cos 35°
Ex. 5 Solve each triangle. Round side lengths to the
nearest tenth and angle measures to the nearest degree.
Y
z
x
35°
Z
Tan Ɵ = opp
adj
Tan 35° = x
10
10Tan 35° = x
x ≈ 7.0
10
X
Cos Ɵ = adj
35° + Y = 90 °
hyp
Y = 55°
Cos 35° = 10
z
z = 10___ ≈ 12.2
Cos 35°