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8-2 Trigonometric Ratio Objectives Find the sine, cosine, and tangent of an acute angle. Use trigonometric ratios to find side lengths in right triangles and to solve real-world problems. Vocabulary trigonometric ratio sine cosine tangent A trigonometric ratio is a ratio of two sides of a right triangle. Relative to A. Relative to B Opposite side is _____ Sin A= Adjacent side is _____ Cos A= Hypotenuse is _____ Tan A= Opposite side is _____ Sin B = Adjacent side is _____ Cos B = Hypotenuse is _____ Tan B = How will you remember your trigonometric ratios? Example 3: Finding Trigonometric Ratios in Special Right Triangles Use a special right triangle to write cos 30° as a fraction. Use a special right triangle to write tan 45° as a fraction. ***8-2 HW Due _____: ______________________________________________________*** 8-3 Solving Right Triangles Objectives Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Example 1: Identifying Angles from Trigonometric Ratios Use the trigonometric ratio to determine which angle of the triangle is A. Use the trigonometric ratio tan A = 1.875 to determine which angle of the triangle is A. If you know the sine, cosine, or tangent of an acute angle measure, you can use the inverse trigonometric functions to find the measure of the angle. Using given measures to find the unknown angle measures or side lengths of a triangle is known as solving a triangle. To solve a right triangle, you need to know two side lengths or one side length and an acute angle measure. Solve for the missing angle. Example 3: Solving Right Triangles Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Example 4: Solving a Right Triangle in the Coordinate Plane The coordinates of the vertices of ∆PQR are P(–3, 3), Q(2, 3), and R(–3, –4). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. Example 5: Travel Application A highway sign warns that a section of road ahead has a 7% grade. To the nearest degree, what angle does the road make with a horizontal line? San Francisco, California, is famous for its steep streets. The steepness of a road is often expressed as a percent grade. Filbert Street, the steepest street in San Francisco, has a 31.5% grade. This means the road rises 31.5 ft over a horizontal distance of 100 ft, which is equivalent to a 17.5° angle. You can use trigonometric ratios to change a percent grade to an angle measure. ***8-3 HW Due _____: ______________________________________________________*** 8-4 Angles of Elevation and Depression An angle of elevation is the angle formed by a horizontal line and a line of sight to a point above the line. In the diagram, 1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. 2 is the angle of depression from the plane to the tower. Since horizontal lines are parallel, 1 2 by the Alternate Interior Angles Theorem. Therefore the angle of elevation from one point is congruent to the angle of depression from the other point. Example 1: Classifying Angles of Elevation and Depression 1 2 4 3 Example 2: Finding Distance by Using Angle of Elevation The Seattle Space Needle casts a 67-meter shadow. If the angle of elevation from the tip of the shadow to the top of the Space Needle is 70º, how tall is the Space Needle? Round to the nearest meter. What if…? Suppose the plane is at an altitude of 3500 ft and the angle of elevation from the airport to the plane is 29°. What is the horizontal distance between the plane and the airport? Round to the nearest foot. Example 3: Finding Distance by Using Angle of Depression An ice climber stands at the edge of a crevasse that is 115 ft wide. The angle of depression from the edge where she stands to the bottom of the opposite side is 52º. How deep is the crevasse at this point? Round to the nearest foot. What if…? Suppose the ranger sees another fire and the angle of depression to the fire is 3°. What is the horizontal distance to this fire? Round to the nearest foot. Example 4: Shipping Application An observer in a lighthouse is 69 ft above the water. He sights two boats in the water directly in front of him. The angle of depression to the nearest boat is 48º. The angle of depression to the other boat is 22º. What is the distance between the two boats? Round to the nearest foot. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot. ***8-4 HW Due _____: ______________________________________________________*** 8-5 Law of Sines and Law of Cosines Example 1: Finding Trigonometric Ratios for Obtuse Angles Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A. tan 103° B. cos 165° C. sin 93° You can use the _____________ of a triangle to find a relationship between the triangle’s side lengths. In ∆ABC, let h represent the length of the altitude from C to From the diagram, Derive Law of Sines sinA=_______ and sinB= _______ You can use the Law of Sines to solve a triangle if you are given… • two angle measures and any side length. ______________ • two side lengths and a non-included angle measure. _________________ Example 2A: Using the Law of Sines Find FG. Round lengths to the nearest tenth. Example B: Using the Law of Sines Find the mQ. Round angle measure to the nearest degree. Example 2C: Using the Law of Sines Find AC. Round lengths to the nearest tenth. ***The Law of Sines cannot be used to solve every triangle*** You can use the Law of Cosines to solve a triangle if you are given… • two side lengths and the included angle measure __________ • three side lengths _________. Example 3A: Using the Law of Cosines Find XZ. Round lengths to the nearest tenth. or Example 3B: Using the Law of Cosines Find the mT. Round angle measure to the nearest degree. Example 4: Sailing Application A sailing club has planned a triangular racecourse, as shown in the diagram. How long is the leg of the race along BC? How many degrees must competitors turn at point C? Round the length to the nearest tenth and the angle measure to the nearest degree. What if…? Another engineer suggested using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. 8-5 HW Due _________________: Pg. 573 (23-25, 29-30, 32-34, 40, 42-44*, 48, 49*, 51-54, 57, 58)