Unit 6 Introduction to Trigonometry Right Triangle Trigonomotry - PowerPoint PPT Presentation

Unit 6 – Introduction to Trigonometry Right Triangle Trigonomotry (Unit 6.1) William (Bill) Finch Mathematics Department Denton High School

Introduction Trig Ratios Solve Right Triangle Applications Summary 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. W. Finch DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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. W. Finch DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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. W. Finch DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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. W. Finch DHS Math Dept Right Triangle Trig 2 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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 . W. Finch DHS Math Dept Right Triangle Trig 3 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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 . W. Finch DHS Math Dept Right Triangle Trig 3 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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 . W. Finch DHS Math Dept Right Triangle Trig 3 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Six Trigonometric Ratios ◮ θ (theta) is an acute angle ◮ ‘opp’ is the side opposite to θ opp hyp ◮ ‘adj’ is the side adjacent to θ ◮ ‘hyp’ is the hypotenuse θ adj sine( θ ) = sin θ = opp cosecant( θ ) = csc θ = hyp hyp opp cosine( θ ) = cos θ = adj secant( θ ) = sec θ = hyp hyp adj tangent( θ ) = tan θ = opp cotangent( θ ) = cot θ = adj adj opp W. Finch DHS Math Dept Right Triangle Trig 4 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Six Trigonometric Ratios ◮ θ (theta) is an acute angle ◮ ‘opp’ is the side opposite to θ opp hyp ◮ ‘adj’ is the side adjacent to θ ◮ ‘hyp’ is the hypotenuse θ adj sine( θ ) = sin θ = opp cosecant( θ ) = csc θ = hyp hyp opp cosine( θ ) = cos θ = adj secant( θ ) = sec θ = hyp hyp adj tangent( θ ) = tan θ = opp cotangent( θ ) = cot θ = adj adj opp W. Finch DHS Math Dept Right Triangle Trig 4 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Reciprocal Functions The cosecant, secant, and cotangent functions are called the reciprocal functions . 1 1 1 csc θ = sec θ = cot θ = sin θ cos θ tan θ W. Finch DHS Math Dept Right Triangle Trig 5 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Example 1 Find exact values for the six trigonometric functions of θ . 24 25 θ 7 W. Finch DHS Math Dept Right Triangle Trig 6 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Example 2 If sin θ = 1 3, find exact values of the five remaining trigonometric functions for the acute angle θ . W. Finch DHS Math Dept Right Triangle Trig 7 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Special Angles 30 ◦ -60 ◦ -90 ◦ Triangle θ 30 ◦ 45 ◦ 60 ◦ 2 x √ √ 60 ◦ x 1 2 3 sin θ 2 2 2 30 ◦ √ √ 3 2 1 cos θ 2 2 2 √ 3 x √ 3 √ tan θ 1 3 45 ◦ -45 ◦ -90 ◦ Triangle 3 √ 2 3 √ csc θ 2 2 3 √ 45 ◦ 2 3 √ √ x sec θ 2 2 2 x 3 √ 45 ◦ √ 3 cot θ 3 1 3 x W. Finch DHS Math Dept Right Triangle Trig 8 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Solve a Right Triangle To solve a right triangle is to find unknown side lengths and/or unknown angles. B a c C A b W. Finch DHS Math Dept Right Triangle Trig 9 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Example 3 Find the value of x . x 7 55 ◦ W. Finch DHS Math Dept Right Triangle Trig 10 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Inverse Trigonometric Functions If sin θ = x , then sin − 1 x = θ . Inverse Sine If cos θ = x , then cos − 1 x = θ . Inverse Cosine If tan θ = x , then tan − 1 x = θ . Inverse Tangent W. Finch DHS Math Dept Right Triangle Trig 11 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Example 4 Use a trigonometric function to find the measure of θ . Round to the nearest degree, if necessary. 15 . 7 12 θ W. Finch DHS Math Dept Right Triangle Trig 12 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Example 5 Solve the right triangle. H 28 f 41 . 4 ◦ G F h W. Finch DHS Math Dept Right Triangle Trig 13 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Example 6 Solve the right triangle. C a 5 B 9 A W. Finch DHS Math Dept Right Triangle Trig 14 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary Angles of Elevation and Depression An angle of elevation is the An angle of depression is the angle formed by a horizontal angle formed by a horizontal line and an observer’s line of line and an observer’s line of sight up to an object. sight down to an object below. Object Observer Depression Elevation Object Observer W. Finch DHS Math Dept Right Triangle Trig 15 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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? W. Finch DHS Math Dept Right Triangle Trig 16 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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? W. Finch DHS Math Dept Right Triangle Trig 17 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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 W. Finch DHS Math Dept Right Triangle Trig 18 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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 W. Finch DHS Math Dept Right Triangle Trig 18 / 18

Introduction Trig Ratios Solve Right Triangle Applications Summary 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 W. Finch DHS Math Dept Right Triangle Trig 18 / 18