Download Complementary Angles and Trigonometric Ratios

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Concept
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Use With Lesson 8-3
T E C H N O L O G Y
Complementary
Angles and
Trigonometric Ratios
Content Standard
G.SRT.7 Explain and use the relationship
between the sine and cosine of complementary
angles.
The acute angles of a right triangle are complementary because the sum of
their measures is 90. There is a relationship between the sine and cosine of
complementary angles.
R
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What is the relationship between the sine and cosine of complementary angles?
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1. Refer to right triangle PQR, shown at the right.
a. How is the opposite side of R related to the adjacent side of Q?
b. How is the adjacent side of R related to the opposite side of Q?
c. Compare sin R and cos Q.
d. Compare sin Q and cos R.
e. Make a conjecture based on your results from parts (c) and (d).
2. Refer to right triangle ABC, shown at the right.
a. Are angles B and C complementary? Justify your answer.
b. Write sine and cosine ratios for angles B and C. What do you notice about the
relationship between sin B and cos C? Between sin C and cos B?
c. Is the statement sin B cos C true for all complementary angles of a right
triangle? Explain.
3. Reasoning Prove that in a right triangle with acute angles B and C,
cos B sin (90 mB).
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hsm12gese_cc04_t05001.ai
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Exercises
4. In the figure at the right, line c is perpendicular to line d.
a. For NLMN, cos L is equal to the sine of which angle? Explain.
b. For NMOP, sin P is equal to the cosine of which angle? Explain.
c. For NMRQ, cos Q is equal to the sine of which angle? Explain.
5. The sine of an acute angle in a right triangle is the ratio of the length of the leg
opposite the angle to the length of the hypotenuse. If you examine the structure of
the word cosine and the relationship of the angles in a right triangle, how do you
think cosine might have gotten its name?
hsm12gese_cc04_t05002.ai
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6. A taut wire runs from the top of a tall pole to the ground. The pole is perpendicular
to the ground. The sine of the angle that the wire makes with the ground is 12
13 .
What is the cosine of the angle that the wire makes with the top of the pole?
Explain.
hsm12gese_cc04_t05003.ai
Concept Byte Parallel Lines and Related Angles
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Complementary Angles and Trigonometric Ratios
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7/29/11 6:08:07 PM
7/29/11 6:09:26 PM