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Example 1 Find Trigonometric Values Example 2 Use One Trigonometric Ratio to Find Another Example 3 Find a Missing Side Length of a Right Triangle Example 4 Solve a Right Triangle Example 5 Find Missing Angle Measures of Right Triangles Example 6 Indirect Measurement Example 7 Use an Angle of Elevation Right Triangles Consider the right triangle ABC in which the measure of acute angle A is identified by the Greek letter theta, θ. The sides of the triangle are the hypotenuse, the leg opposite θ, and the leg adjacent to θ. Right Triangles Using these sides, you can define six trigonometric functions: Sine (sin) Cosine (cos) Tangent (tan) Secant (sec) Cosecant (csc) Cotangent (cot) How can YOU remember SOHCAHTOA? Special Right Triangles 45 – 45 – 90 30 – 60 – 90 Introduction to Trigonometric Ratios with Special Right Triangles Packet Pages 2 – 4 Multiple-Choice Test Item If find the value of csc A. A B C D Read the Test Item Draw a right triangle and label one acute angle A. Since and , label the opposite leg 5 and the adjacent leg 3. Solve the Test Item Use the Pythagorean Theorem to find c. Pythagorean Theorem Replace a with 3 and b with 5. Simplify. Take the square root of each side. Now find csc A. Cosecant ratio Replace hyp with and opp with 5. Answer: D Multiple-Choice Test Item If find the value of cos B. A B C D Answer: C Write an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest tenth. The measure of the hypotenuse is 12. The side with the missing length is opposite the angle measuring 60 . The trigonometric function relating the opposite side of a right triangle and the hypotenuse is the sine function. Sine ratio Replace with 60 , opp with x, and hyp with 12. Multiply each side by 12. Answer: The value of x is or about 10.4. Write an equation involving sin, cos, or tan that can be used to find the value of x. Then solve the equation. Round to the nearest tenth. Answer: or about 8.7 Solving a General Right Triangle If you know the measures of any two sides of a right triangle or the measures of one side and one acute angle, you can determine the measures of all the sides and angles of the triangle. This process of finding the missing measures is known as solving a right triangle. Solve XYZ. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. x z You know the measures of one side, one acute angle, and the right angle. You need to find x, z, and Y. Find x and z. Multiply each side by 11. Use a calculator. x Multiply each side by 11. Use a calculator. z Find Y. Angles X and Y are complementary. Solve for Y. Answer: Therefore, , , and . Solve XYZ. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. Answer: Using the Inverse • You can use the inverse capabilities to find the measure of an angle when one of its trigonometric ratios is known. Solve ABC. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. You know the measures of the sides. You need to find A and B. Find A. Use a calculator and the SIN–1 function to find the angle whose sine is Keystrokes: 2nd [SIN–1] 9 17 ) ENTER 31.96571875 Find B. Angles A and B are complementary. Solve for B. Answer: Therefore, and . Solve ABC. Round measures of sides to the nearest tenth and measures of angles to the nearest degree. Answer: Uses Trigonometry has many practical applications. Among the most important is the ability to find distances or lengths that either cannot be measured directly or are not easily measured directly. Bridge Construction In order to construct a bridge across a river, the width of the river must be determined. A stake is planted on one side of the river directly across from a second stake on the opposite side. At a distance 30 meters to the left of the stake, an angle of 55 is measured between the two stakes. Find the width of the river. Let w represent the width of the river. Write an equation using a trigonometric function that involves the ratio of w and 30. Multiply each side by 30. Use a calculator. Answer: The width is about 42.8 meters. Bridge Construction To construct a bridge, a stake is planted on one side of the river directly across from a second stake on the opposite side. The angle between the two stakes is measured at a distance 20 meters away from the stake, and found to be 50 . Find the width of the river. Answer: about 23.8 meters Skiing A run has an angle of elevation of 15.7 and a vertical drop of 1800 feet. Estimate the length of this run. Let represent the length of the run. Write an equation using a trigonometric function that involves the ratio of and 1800. Solve for Use a calculator. Answer: The length of the run is about 6652 feet. Skiing A run has an angle of elevation of 23 and a vertical drop of 1000 feet. Estimate the length of this run. Answer: about 2559 feet Summary • What does SOHCAHTOA stand for? • What is the minimum information you have to have about a right triangle to solve it? • Assignment: finish packet pages 2-4 and problems 29-40 on page 707 in the textbook. Click the mouse button or press the Space Bar to display the answers.