Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name: __________________________________ Period _____ Right Triangle Trigonometry Assignment: Section 12-1 #1-7, 11, 13, 14, and 16 Calculator Note: Use DEGREE MODE Important Vocabulary you need to know: (1) Corresponding Sides & Corresponding Angles of Similar Triangles. (2) Right Triangle (3) Adjacent side to an angle (4) Opposite side to an angle (5) Hypotenuse the longest side in a right triangle (6) Ratio is a fraction Side a is opposite angle A (7) Trigonometric Ratio: Sine, Cosine and Tangent Side b is adjacent to angle B Pythagorean Theorem: a 2 b 2 c 2 Similar Triangles: You’ve learned that corresponding sides of similar triangles have the same ratio. The three triangles shown below are similar because all three corresponding angles are congruent. The ratio of the length of the shorter side to the length of the middle-length side is always 0.75. shortest leg 6 4.8 3 0.75 middle leg 8 6.4 4 (1) Find the ratio of the length of the middle-length side to the longest side: middle-length leg longest leg ______ (2) Find the ratio of the length of the shortest side to the longest side: shortest leg longest leg ______ These ratios are called trigonometric ratios (abbreviated as “trig ratios”). For any acute angle A in a right triangle, the three trig ratios are defined: Opposite leg a Hypotenuse c Adjacent leg b Cosine of A is a ratio: cos( A) Hypotenuse c Opposite leg a Tangent of A is a ratio: tan( A) Adjacent leg b Sine of A is a ratio: sin( A) Acute angles are A & B (3) Write the Sine, Cosine and Tangent of angle B using the letters a, b, and c. Sine of B is a ratio: sin( B) Opposite leg Hypotenuse Cosine of B is a ratio: cos( B) Adjacent leg Hypotenuse Tangent of B is a ratio: tan( B) Opposite leg Adjacent leg You can use the trig ratios to find an unknown side in a right triangle. Example: Find the unknown length x in the triangle shown below. Label the hypotenuse and the opposite & adjacent sides to a known angle. (see diagram) Write down the trig ratio you need to use. Adjacent leg Hypotenuse cos( A) Fill in the numbers and solve for the unknown side. cos(43) 20 x Solve for x. ( x) cos(43) 20 20 x 27.35 cm cos(43) Hint: You use the cosine ratio on this example because you know the adjacent side and want to find the hypotenuse. You try this one: Find the unknown length x in the triangle shown below. Label the hypotenuse and the opposite & adjacent sides to a known angle ON THE DIAGRAM Write down the trig ratio you need to use. Fill in the numbers and solve for the unknown side. Solve for x. Hint: You use the sine ratio on this example because you know the opposite side and want to find the hypotenuse. You try this another one: Find the unknown length x Label the hypotenuse and the opposite & adjacent sides to a known angle ON THE DIAGRAM Write down the trig ratio you need to use. Fill in the numbers and solve for the unknown side. Solve for x. Hint: You use the tangent ratio on this example because you know the adjacent side and want to find the opposite. You try this another one: Find the unknown length x Label the hypotenuse and the opposite & adjacent sides to a known angle ON THE DIAGRAM Write down the trig ratio you need to use. Fill in the numbers and solve for the unknown side. Solve for x. Hint: You use the tangent ratio on this example because you know the opposite side and want to find the adjacent. You try this another one: Find the unknown length x Label the hypotenuse and the opposite & adjacent sides to a known angle ON THE DIAGRAM Write down the trig ratio you need to use. Fill in the numbers and solve for the unknown side. Solve for x. Hint: You use the sine ratio on this example because you know the hypotenuse and want to find the opposite side. You can use the trig ratios to find an unknown angle in a right triangle. Remember that the inverse function switches the x and y values. That means if you have a trig ratio such as: y cos( angle) the inverse trig function would be angle cos 1 1 To summarize: USE inverse trig functions ( cos , sin 1 y A, tan 1 A ) to find an angles. Example: Find the unknown angle x Label the known sides as hypotenuse and adjacent to the angle you are trying to find. (see the diagram) Write down the trig ratio you need to use. Adjacent leg cos( A) Hypotenuse Fill in the value for the sides that you know. 20 cos( x) 75 Use the inverse cosine function to solve for the unknown 20 angle x: x cos 1 75 Use your calculator to get an approximate answer. 20 x cos 1 74.5 degrees 75 You try this one: Find the unknown angle x Label the known sides as hypotenuse, adjacent or opposite side to the angle you are trying to find ON THE DIAGRAM. Write down the trig ratio you need to use. Fill in the value for the sides that you know. Use the inverse trig function to solve for the unknown angle x: Use your calculator to get an approximate answer. Hint: You use the cosine ratio on this example because you know the adjacent side and the hypotenuse. Hint: You use the tangent ratio on this example because you know the opposite side and adjacent side. You try this one: Find the unknown angle x Label the known sides as hypotenuse, adjacent or opposite side to the angle you are trying to find ON THE DIAGRAM. Write down the trig ratio you need to use. Fill in the value for the sides that you know. Use the inverse trig function to solve for the unknown angle x: Hint: You use the sine ratio on this example because you know the opposite side and hypotenuse. Use your calculator to get an approximate answer. When you are working with the trig ratios always draw the triangle and label the sides as either: (1) opposite or (2) adjacent to an angle and the hypotenuse. If you do this it will help you select the correct trig ratio to use. Directions: Draw a right triangle for each problem. Label the sides and angle, and then solve to find the unknown measure. Example: cos(40 ) a adjacent 20 hypotenuse Draw a triangle: label hypotenuse & adjacent You try: tan(40 ) b 5 Draw a triangle and label the diagram like the example shown at the left. Write the trig ratio and solve for the unknown side. a cos(40 ) 20 a 20cos(40 ) 15.3 Write the trig ratio and solve for the unknown side. Example: Use trig ratios to solve problems. Two hikers leave their campsite. Tom walks north 5.7 miles and Sam walks east 4.2 miles. (a) After Tom gets to his destination, he looks directly toward Sam’s destination. What is the measure of the angle between the path Tom walked and his line of sight to Sam’s destination? (b) How far apart are Sam and Tom? Draw a right triangle using the given information. Select the trig ratio you need to use and write in the numbers: tan( A) Opposite 4.2 Adjacent 5.7 Use the inverse tangent function to solve for the angle. 4.2 A tan 1 36.4 degrees 5.7 Use Pythagorean Theorem or the sine or cosine ratio to solve for the distance, D. sin(36.4) 4.2 D 4.2 sin(36.4) D 7.1 miles D cos(36.4) 5.7 D 5.7 cos(36.4) D 7.1 miles D ( Sam) 2 (Tom) 2 D 2 (4.2) 2 (5.7) 2 D 2 50.13 D 2 D 50.13 D 7.1 miles Answers: (a) 36.4 degrees (b) 7.1 miles YOU TRY: Two hikers leave their campsite. Sue walks south 3.2 miles and Kate walks west 4.7 miles. (a) After Sue gets to her destination, she looks directly toward Kate’s destination. What is the measure of the angle between the path Sue walked and her line of sight to Kate’s destination? (b) How far apart are Sue and Kate? Select the trig ratio you need to use and write in the numbers: Draw a right triangle using the given information. Use the inverse tangent function to solve for the angle. Use Pythagorean Theorem or the sine or cosine ratio to solve for the distance, D. Use sine ratio Use cosine ratio Use Pythagorean Theorem Answers: (a) _________ degrees (b) __________ miles