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4.1 – Right Triangle Trigonometry

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Sin Ɵ = opp hyp

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Sin Ɵ = opp hyp Csc Ɵ = hyp opp

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Csc Ɵ = hyp opp

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Csc Ɵ = hyp opp Sec Ɵ = hyp adj

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp Csc = 1 sin

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp Csc = 1 sin Sec = 1 cos

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Sin Ɵ = opp hyp Cos Ɵ = adj hyp Tan Ɵ opp adj Csc Ɵ = hyp opp Sec Ɵ = hyp adj Cot Ɵ = adj opp Csc = 1 sin Sec = 1 cos Cot = 1 tan

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp hyp Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 hyp 17 Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp hyp 17 opp Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj hyp Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15 hyp 17 Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp hyp 17 adj Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp adj Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp = 8 adj 15 Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp = 8 Cot Ɵ = adj adj 15 opp Ɵ 8 17 15

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Ex. 1 Find the exact values of the six trigonometric functions of Ɵ. Sin Ɵ = opp = 8 Csc Ɵ = hyp = 17 hyp 17 opp 8 Cos Ɵ = adj = 15Sec Ɵ = hyp = 17 hyp 17 adj 15 Tan Ɵ = opp = 8 Cot Ɵ = adj = 15 adj 15 opp 8 Ɵ 8 17 15

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ.

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Ɵ

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Ɵ 2

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Ɵ 2 5

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Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp hyp Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp = √21 hyp 5 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp adj Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp opp Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ opp √21 Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 hyp 5 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp hyp 5 adj Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 adj 2 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 Cot Ɵ = adj adj 2 opp Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 Cot Ɵ = adj = 2_ adj 2 opp √21 Ɵ √21 2 5

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Ex. 2 If Cos Ɵ = 2 / 5, find the exact values of the five remaining trigonometric functions for Ɵ. Csc Ɵ = hyp = 5_ = 5√21 opp √21 21 Sin Ɵ = opp = √21 Sec Ɵ = hyp = 5 hyp 5 adj 2 Tan Ɵ = opp = √21 Cot Ɵ = adj = 2_ = 2√21 adj 2 opp √21 21 Ɵ √21 2 5

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Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. 42 18 x

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Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp 42 18 x

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Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp Cos 42° = x 18 42 18 x

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Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp Cos 42° = x 18 18Cos 42° = x 42 18 x

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Ex. 3 Find the value of x. Round to the nearest tenth, if necessary. Cos Ɵ = adj hyp Cos 42° = x 18 18Cos 42° = x 13.4 ≈ x 42 18 x

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Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Ɵ 26 11

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Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Ɵ 26 11

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Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Tan Ɵ = 26 11 Ɵ 26 11

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Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Tan Ɵ = 26 11 Ɵ = tan -1 ( 26 / 11 ) Ɵ 26 11

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Ex. 4 Find the measure of angle Ɵ. Round to the nearest degree, if necessary. Tan Ɵ = opp adj Tan Ɵ = 26 11 Ɵ = tan -1 ( 26 / 11 ) Ɵ ≈ 67° Ɵ 26 11

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj Tan 35° = x 10 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj Tan 35° = x 10 10Tan 35° = x 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = opp adj Tan 35° = x 10 10Tan 35° = x x ≈ 7.0 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x 10 10Tan 35° = x x ≈ 7.0 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = x x ≈ 7.0 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ ≈ 12.2 x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj35° + Y = 90 ° adj hyp Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ ≈ 12.2 x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

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Ex. 5 Solve each triangle. Round side lengths to the nearest tenth and angle measures to the nearest degree. Tan Ɵ = oppCos Ɵ = adj35° + Y = 90 ° adj hypY = 55° Tan 35° = x Cos 35° = 10 10 z 10Tan 35° = xz = 10___ ≈ 12.2 x ≈ 7.0 Cos 35° 35° x 10 z Y ZX

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