Ch 8: Right Triangles and Trigonometry 8-1 The Pythagorean Theorem and Its Converse 8-2 Special Right Triangles 8-3 The Tangent Ratio

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1 Ch 8: Right Triangles and Trigonometry 8-1 The Pythagorean Theorem and Its Converse 8- Special Right Triangles 8-3 The Tangent Ratio 8-1: The Pythagorean Theorem and Its Converse Focused Learning Target: I will be able to Use the Pythagorean Theorem Use the Converse of the Pythagorean Theorem. Vocabulary: Pythagorean Triple 8-4 Sine and Cosine Ratios 8-5 Angles of Elevation and Depression CA Standard(s): Geo 15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. Although the sides of a right triangle will always satisfy the Pythagorean Theorem, Pythagorean triples are a special set of positive whole numbers that satisfy the equation a + b = c. Examples of common Pythagorean Triples: Not Pythagorean Triples: 3, 4, 5 8, 15, 17 7, 10, ,, 4 5, 1, 13 7, 4, 5-3,-4, 5 1,, Finding the length of a side using the Pythagorean Theorem and determining if it forms a Pythagorean triple: Find the missing side, and then determine if it forms a Pythagorean triple with the other two sides. We ll do one together: Find the missing side(s), and then determine if all three sides form a Pythagorean triple. 1

2 You Try: Find the missing side(s) and determine if it forms a Pythagorean triple with the other two sides. Simplifying Radicals: Using Simplest Radical Form: When solving for a variable by taking the square root, the value is often not a perfect square. In those cases it is often best to leave your answer as a radical in simplest form. Here is one method: 1) Divide the number by, 3, 5, 6, etc (skip the perfect squares) ) When the result is a perfect square you can simplify by taking the square of the perfect square. Leave the other factor inside the radical. If it does not factor in this way, then it is already in simplest radical form. We ll do one: You try one: Simplify: Simplify: Simplify: The first 0 Perfect Squares chart 1 = 1 11 = 11 = 4 1 = = 9 13 = = = = 5 15 = 5 6 = = 56 7 = = 89 8 = = 34 9 = = = = 400 Examples: Using the Pythagorean Theorem and Simplifying Radicals If a =, b = 4, find c. Leave your answer in simplest radical form.

3 We ll do one together: Find x. Leave your answer in simplest radical form. You Try: If a = 6, c = 1, find b. Leave your answer in simplest radical form. Examples: Applying the Pythagorean Theorem in real-world situations The Parks Department rents Paddle boats at docks near each entrance to the park. To the nearest meter, how far is it to paddle from one dock to the other? We ll do one together: A highway detour affects a company s delivery route. The plan showing the old route and the detour is below. How many extra miles will the trucks travel once the detour is established? Round your answer to the nearest tenth of a mile. 3

4 You Try: A bike messenger has just been asked to make an additional stop. Instead of biking straight from the law office to the court, she is going to stop at City Hall in between. Using the picture below, determine how many additional miles she will need to travel. Leave your answer in simplest radical form. Using the converse to determine if it is a right triangle: Is the triangle below a right triangle? Explain. We ll do one together: Is the triangle below a right triangle? Explain. You Try: Is the triangle below a right triangle? Explain. 4

5 8- Special Right Triangles Focused Learning Target: I will be able to Use the properties of triangles. Use the properties of triangles. CA Standard(s): Geo 15.0: Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles. Geo 0.0: Students know and are able to use angle and side relationships in problems with special right triangles, such as 30, 60, 90 triangles and 45, 45, and 90 triangles. There are two special triangles that are often encountered. In this section, we will explore the characteristics that are unique to these triangles. The triangle is an isosceles right triangle. It has all of the properties of an isosceles triangle and all of the properties of a right triangle. The theorem below is often used as a quicker way to find missing lengths of these special right triangles compared to the Pythagorean Theorem. Finding the hypotenuse We ll do one together: You try: Find the value of the variable: Find the value of the variable: Find the value of the variable: Finding the length of a leg: Find the length of a leg of a Triangle with a hypotenuse of length 10. 5

6 We ll do one together: Find the value of the variables: You try: Find the value of the variable: Another type of special triangle is the triangle for which we use the theorem below: Find the lengths of the missing sides. We ll do one together: Find the missing variables: 6

7 You Try: Find the missing variables: Some Real-World Situations The moose warning sign below is an equilateral triangle. It is one meter high. Find the lengths of each of the sides. We ll do one: After heavy winds damaged a farmhouse, workers placed a 6-m brace against its side at a 45 angle. Then, at the same spot on the ground, they placed a second brace at 30 angle. a) How long is the longer brace? Round your answer to the nearest tenth of a meter. b) About how much higher on the house does the longer brace reach than the shorter brace? 7

8 You try: Jefferson Park sits on one square city block 300 ft on each side. Sidewalks connect the opposite corners. a) Draw the diagram b) About how long is each diagonal sidewalk? 8-3 The Tangent Ratio Focused Learning Target: I will be able to: Use the tangent ratios to determine side lengths in triangles CA Standard(s): Geo 18.0: students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. Geo 19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Vocabulary: Tangent: the tangent of acute A in a right triangle is the ratio of the length of the leg opposite of A to the length of the leg adjacent to A. Example 1: Writing tangent ratios: We ll do one: You try: Write the tangent ratios for T and U. Write the tangent ratios for Eand F. 8

9 Example : Real-World-Connection We ll do one: Find the distance from the boathouse on shore to the cabin on the island. You try: Find x to the nearest whole number. 1 opp Example 3: Using the Inverse of Tangent, tan, to find the missing angle adj We ll do one: You Try: Find the measure of angle X to the Find the value of x to the nearest Find the value of x to the nearest nearest degree. degree. degree. 9

10 8-4 Sine and Cosine Ratios Focused Learning Target: Use sine and cosine to determine side lengths in triangles CA Standard(s): Geo 18.0: Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They are able to use elementary relationships between them. Geo 19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. Vocabulary: Example 1: Writing the Sine and Cosine Ratios We ll do one: Write the cosine and sine ratios for X and Y. You Try: Write the ratios for sin P and cos Q. 10

11 How can we remember the ratios of Sine, Cosine, and Tangent? SOH, CAH, TOA!!!! Opp O Sin ( angle) = S = SOH Hyp H Adj A Cos ( angle) = C = CAH Hyp H Opp O Tan ( angle) = T = TOA Adj A Example : Real-World Connection We ll do one: Find the value of x. Round answer to the nearest tenth. You Try: Find the value of x. Round answer to the nearest tenth. 11

12 1 opp 1 adj Example 3: Using the Inverse of Sine, sin and Cosine, cos hyp hyp Find the value of x. Round your answer to the nearest degree. We ll do one: Find the value of x. Round your answer to the nearest degree. You try: Find the value of x. Round your answer to the nearest degree. 1

13 8-5 Angles of Elevation and Depression Focused Learning Target: Use angles of elevation and depression to solve problems. Vocabulary: Angle of elevation Angle of depression CA Standard(s): Geo 18.0: Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. Geo 19.0: Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. The angle of elevation is the angle formed by taking the horizontal line (as if you re looking straight ahead) and then looking upwards. The angle of depression is the angle formed by taking the horizontal line and then looking downwards. In the picture to the right, notice that the angle of depression from the person in the hot air balloon is congruent to the angle of elevation from the person on the ground. Why are those angles congruent? Example 1: Identifying angles of elevation and depression If you were at the peak of the mountain at 1, what type of angle would that be; an angle of elevation or depression? We ll try some together: What about at? 3? You try: What about at 4? 13

14 Example : Finding lengths in real-world scenarios using trigonometry You are hiking and you spot a rock climber on a nearby cliff at a 3 o angle of elevation. The horizontal ground distance to the base of the cliff is 1000 ft. Find the height of the rock climber. (hint: always try to draw the situation if none is given) We ll do one together: To approach runway 17 of the Ponca City Municipal Airport in Oklahoma, the pilot must begin a 3 o descent starting from an altitude of 714 ft. The airport s altitude is 1007 ft above sea level. How many feet from the runway is the airplane at the start of this approach? Then convert your answer into miles. You try: A coast guard helicopter pilot sights a life raft at a 6 o angle of depression. The helicopter s altitude is 3 km. What is the helicopter s horizontal distance from the raft? 14