The Sine and Cosine Ratios

  1. In the Gizmotm, on the Sine tab, set mangleA to 25. (To quickly set a value, type a number in the box to the right of the slider and press ENTER.) Notice triangleABC.
    1. Which angle has degree measure 25? Which angle is the right angle? How can you quickly calculate mangleB? What is mangleB?
    2. Which side is the hypotenuse? Which leg of triangleABC is opposite angleA? Which leg is adjacent to angleA?
    3. Click on Click to measure lengths and use the interactive rulers to find the lengths of the side opposite angleA and the hypotenuse. (For help using the rulers, click on Gizmo help, below the Gizmo.) What are those lengths? Click on Show side lengths to check your work.
    4. In a right triangle, the ratio of the length of the leg opposite an angle to the length of the hypotenuse is called the sine ratio of a given angle. (This is often shortened to "Sine equals opposite over hypotenuse.") The sine ratio is abbreviated sin. What is sin 25°? State your answer as a fraction. Then use a calculator to do the division. Round your answer to three decimal places. Click on Show sine computation to check your answer.
    5. Reshape triangleABC by dragging C, keeping mangleA = 25. What is always true about the ratio of BC to AB? Is sin 25° a constant, regardless of the size of the right triangle? Explain why or why not.
  2. With Show side lengths turned on, turn off Show sine computation and rulers. Set mangleA = 58.
    1. To find sin 58°, which sides of the triangle do you need to know? Write the ratio you will use to calculate sin 58°. Which side length is in the numerator? Which side length is in the denominator?
    2. Use a calculator to divide the numerator by the denominator. Round your answer to three decimal places. What is sin 58°? Use the Gizmo to check your answer.
  3. Vary mangleA using the slider. Observe the value of sin A.
    1. As mangleA increases from 0 to 90, what happens to sin A? What happens as mangleA decreases from 90 to 0? Which is greater, sin 10° or sin 80°?
    2. Describe the sine ratio of angleA when mangleA = 0. What is sin 0°? Describe the sine ratio of angleA when mangleA = 90. What is sin 90°?
    3. What is the greatest possible value of sin A? What is the least possible value? Explain.

The cosine ratio

  1. Click on the Cosine tab. With Show side lengths turned on, turn off rulers and set mangleA to 65.
    1. Which leg is adjacent to angleA? Which side is the hypotenuse? What are the lengths of these two sides?
    2. In a right triangle, the ratio of the length of the leg adjacent an angle to the length of the hypotenuse is called the cosine ratio of a given angle. (This is often shortened to "Cosine equals adjacent over hypotenuse.") The cosine ratio is abbreviated cos. What is cos 65°? State your answer as a fraction. Then use a calculator to do the division. Round your answer to three decimal places. Click on Show cosine computation to check your answer.
    3. Drag C to form other triangles with mangleA = 65. What do you notice about the ratio of AC to AB? Is cos 65° a constant, regardless of the size of the right triangle? Explain.
  2. Vary mangleA by moving the slider. Observe the value of cos A.
    1. As mangleA increases from 0 to 90, what happens to cos A? What happens as mangleA decreases from 90 to 0? Which is greater, cos 10° or cos 80°?
    2. Describe the cosine ratio of angleA when mangleA = 0. What is cos 0°? Describe the cosine ratio of angleA when mangleA = 90. What is cos 90°?
    3. What is the greatest possible value of cos A? What is the least possible value? Explain.

Real-world problems

Use the Gizmo and a calculator to solve the problems below.

  1. A 14-foot ladder is placed against a building. The ladder makes an angle of 50° with the ground. How high up on the side of the building does the ladder reach?
    1. Draw a diagram illustrating this problem. Label angles and sides.
    2. Should you use sine or cosine to solve this problem? Why? Write an equation that will solve the problem.
    3. What is sin 50°? Substitute that value in your equation. How do you solve for the variable? Solve the equation. How high up on the building will the ladder reach? Check your answer using the Gizmo.
  2. You move the 14-foot ladder closer so that it now makes a 60° angle with the ground. How close is the base of the ladder to the building?
    1. Draw a diagram illustrating this problem and label angles and sides.
    2. Write an equation and solve it. What is your answer?
  3. If you move the ladder so that it is at the steepest safe angle for working, the distance from the building will be one-third of the length of the ladder. Make a sketch and label the sides. What is the measure of the angle that the ladder makes with the ground? Use the inverse of sine (arcsin or sin−1) or the inverse of cosine (arccos or cos−1) on a calculator to find the answer.