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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°