6. The hypotenuse of a triangle is 24.2 ft. Explain how to find the lengths of the legs of the triangle. 1. In the center of town there is a square park with side length 30 ft. If a person walks from one corner of the park to the opposite corner, how far does the person walk? Round to the nearest foot. A. 21 ft B. 42 ft C. 52 ftD. 60 ft 5. A sailing course is in the shape of an equilateral triangle. If the course has an altitude of 9 mi, what is the perimeter of the triangle? page 360 Lesson 10-2 | Special Right Triangles

What is the ratio of the length of the shorter leg to the length of the hypotenuse for each of  ADF,  AEG, and  ABC? Make a conjecture based on your results. Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems.

SOH-CAH-TOA Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems.

In a right triangle, there are actually six possible trigonometric ratios, or functions. A Greek letter (such as theta,  or beta,  ) will now be used to represent the angle.  Notice that the three new ratios at the right are reciprocals of the ratios on the left. Applying a little algebra shows the connection between these functions.

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems.

What are the sine, cosine, and tangent ratios for  G? Writing Trigonometric Ratios SOH-CAH-TOA Using a Trigonometric Ratio to Find Distance For parts (a)–(c), find the value of w to the nearest tenth.

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Using a Trigonometric Ratio to Find Distance d. A section of Filbert Street in San Francisco rises at an angle of about 17. If you walk 150 ft up this section, what is your vertical rise? Round to the nearest foot. 17  ┌ Filbert Street 150 ft x SOH-CAH-TOA

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Using Inverses a. Use the figure below. What is m  Y to the nearest degree? SOH-CAH-TOA b. Suppose you know the lengths of all three sides of a right triangle. Does it matter which trigonometric ratio you use to find the measure of any of the three angles? Explain. No, No, you can use any trig. ratio as long as you identify the appropriate opp. or adj. leg to each acute angle. you can use any trig. ratio as long as you identify the appropriate opp. or adj. leg to each acute angle.

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Use the figure below. What is m  A to the nearest degree? SOH-CAH-TOA

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Do you know HOW? 1. What is the value of x ? Round to the nearest tenth. 2. One angle of a rhombus measures 70. The diagonal of the rhombus bisecting this angle measures 10 cm. What is the perimeter of the rhombus to the nearest tenth of a centimeter? SOH-CAH-TOA  70  35  ┐ 5 5 P = 4 x x 10 Solve for x: Solve for perimeter, P: 

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Do you know HOW? 3. The height of a local cell tower is 75 ft. A surveyor measures the length of the tower’s shadow cast by the sun as 36 ft. What is the measure of the angle the sun’s ray makes with the ground? Round to the nearest degree. ┐ 75 ft  36 ft Let  = the angle the sun’s rays make with the ground

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Do you understand? 4. Vocabulary Some people use SOH-CAH-TOA to remember the trigonometric ratios for sine, cosine, and tangent. Why do you think that word might help? (Hint: Think of the first letters of the ratios.)

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Do you understand? 5. Connect Mathematical Ideas (1)(F) A student states that sin A > sin X because the lengths of the sides of  ABC are greater than the lengths of the sides of  XYZ. Do you agree with the student? Provide mathematical proof to defend your position. Proof: and  ABC   XYZ by AA  Postulate, because ratios of corresponding sides are proportional.

Objective: Determine the lengths of sides and measures of angles in a right  by applying the trigonometric ratios sine, cosine, and tangent to solve problems. Do you understand? YES, only if the right triangle is isosceles. 6. Justify Mathematical Arguments (1)(G) Is it possible for the cosine of an acute angle in a right triangle to be equal to its sine? If so, provide proof. If not, explain why it is not possible. ┐ A B C  Proof: