Unit 6 Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1) William (Bill) Finch Mathematics Department Denton High School
Lesson Goals When you have completed this lesson you will: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Right Triangle Trig 2 / 18
Lesson Goals When you have completed this lesson you will: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Right Triangle Trig 2 / 18
Lesson Goals When you have completed this lesson you will: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Right Triangle Trig 2 / 18
Lesson Goals When you have completed this lesson you will: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Right Triangle Trig 2 / 18
Trigonometry The word trigonometry comes from the Greek language for measurement of triangles. The development of physics and calculus in the 16th-17th centuries led to viewing trigonometric relationships as functions with real numbers as their domains. We now study and apply trigonometry concepts using both triangles and circles. Right Triangle Trig 3 / 18
Trigonometry The word trigonometry comes from the Greek language for measurement of triangles. The development of physics and calculus in the 16th-17th centuries led to viewing trigonometric relationships as functions with real numbers as their domains. We now study and apply trigonometry concepts using both triangles and circles. Right Triangle Trig 3 / 18
Trigonometry The word trigonometry comes from the Greek language for measurement of triangles. The development of physics and calculus in the 16th-17th centuries led to viewing trigonometric relationships as functions with real numbers as their domains. We now study and apply trigonometry concepts using both triangles and circles. Right Triangle Trig 3 / 18
Six Trigonometric Ratios θ (theta) is an acute angle opp is the side opposite to θ adj is the side adjacent to θ hyp is the hypotenuse sine(θ) = sin θ = opp hyp cosine(θ) = cos θ = adj hyp tangent(θ) = tan θ = opp adj opp hyp θ adj cosecant(θ) = csc θ = hyp opp secant(θ) = sec θ = hyp adj cotangent(θ) = cot θ = adj opp Right Triangle Trig 4 / 18
Six Trigonometric Ratios θ (theta) is an acute angle opp is the side opposite to θ adj is the side adjacent to θ hyp is the hypotenuse sine(θ) = sin θ = opp hyp cosine(θ) = cos θ = adj hyp tangent(θ) = tan θ = opp adj opp hyp θ adj cosecant(θ) = csc θ = hyp opp secant(θ) = sec θ = hyp adj cotangent(θ) = cot θ = adj opp Right Triangle Trig 4 / 18
Reciprocal Functions The cosecant, secant, and cotangent functions are called the reciprocal functions. csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ Right Triangle Trig 5 / 18
Example 1 Find exact values for the six trigonometric functions of θ. 24 25 7 θ Right Triangle Trig 6 / 18
Example 2 If sin θ = 1, find exact values of the five remaining 3 trigonometric functions for the acute angle θ. Right Triangle Trig 7 / 18
Special Angles 30-60 -90 Triangle x 60 2x 30 3x 45-45 -90 Triangle 2x 45 45 x x θ 30 45 60 sin θ 1 2 2 2 3 2 2 2 3 1 3 3 2 1 cos θ 2 tan θ 3 csc θ 2 2 2 3 3 2 sec θ 3 3 2 2 cot θ 3 1 3 3 Right Triangle Trig 8 / 18
Solve a Right Triangle To solve a right triangle is to find unknown side lengths and/or unknown angles. B a c C b A Right Triangle Trig 9 / 18
Example 3 Find the value of x. x 7 55 Right Triangle Trig 10 / 18
Inverse Trigonometric Functions Inverse Sine If sin θ = x, then sin 1 x = θ. Inverse Cosine If cos θ = x, then cos 1 x = θ. Inverse Tangent If tan θ = x, then tan 1 x = θ. Right Triangle Trig 11 / 18
Example 4 Use a trigonometric function to find the measure of θ. Round to the nearest degree, if necessary. 15.7 θ 12 Right Triangle Trig 12 / 18
Example 5 Solve the right triangle. H f 28 41.4 G h F Right Triangle Trig 13 / 18
Example 6 Solve the right triangle. C a 5 B 9 A Right Triangle Trig 14 / 18
Angles of Elevation and Depression An angle of elevation is the angle formed by a horizontal line and an observer s line of sight up to an object. An angle of depression is the angle formed by a horizontal line and an observer s line of sight down to an object below. Object Observer Depression Observer Elevation Object Right Triangle Trig 15 / 18
Example 7 Split Rock Lighthouse has stood on the north shore of Lake Superior since 1909. When first lit in 1910 the light could be seen from up to 35 km (a little over 20 miles). The lighthouse is 16 m tall and sits atop a cliff that is 40 m. If a boat was on the lake at a distance of 35 km from the lighthouse, what would be the angle of depression from the top of the lighthouse? Right Triangle Trig 16 / 18
Example 8 At a point 300 feet from the base of the CN Tower the angle of elevation up to the SkyPod (once the worlds highest public observation deck) is 78.4 and the angle of elevation to the top of the tower is 80.6. How much higher above the SkyPod is the top of the tower? Right Triangle Trig 17 / 18
What You Learned You can now: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27, 31-35 odd, 39-45 odd, 49, 53 Right Triangle Trig 18 / 18
What You Learned You can now: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27, 31-35 odd, 39-45 odd, 49, 53 Right Triangle Trig 18 / 18
What You Learned You can now: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27, 31-35 odd, 39-45 odd, 49, 53 Right Triangle Trig 18 / 18
What You Learned You can now: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27, 31-35 odd, 39-45 odd, 49, 53 Right Triangle Trig 18 / 18
What You Learned You can now: Find values of trigonometric functions for acute angles of right triangles. Solve right triangles. Apply right triangle trigonometry to model real-world applications. Do problems Chap 4.1 #1, 5, 11, 13, 17, 21, 25, 27, 31-35 odd, 39-45 odd, 49, 53 Right Triangle Trig 18 / 18