Download 7.3 Sine and Cosine of Complementary Angles

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Chapter 7. Trigonometry
7.3 Sine and Cosine of Complementary Angles
Here you will explore how the sine and cosine of complementary angles are related.
∆ABC is a right triangle with m6 C = 90◦ and sin A = k. What is cos B?
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Guidance
Recall that the sine and cosine of angles are ratios of pairs of sides in right triangles.
• The sine of an angle in a right triangle is the ratio of the side opposite the angle to the hypotenuse.
• The cosine of an angle in a right triangle is the ratio of the side adjacent to the angle to the hypotenuse.
In the examples, you will explore how the sine and cosine of the angles in a right triangle are related.
Example A
Consider the right triangle below. Find the sine and cosine of angles A and B in terms of a, b, and c. What do you
notice?
Solution: sin A = ac , sin B = bc , cos A = bc , cos B = ac . Note that sin A = cos B and sin B = cos A.
Example B
Consider the triangle from Example A. How is 6 A related to 6 B?
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7.3. Sine and Cosine of Complementary Angles
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Solution: The sum of the measures of the three angles in a triangle is 180◦ . This means that m6 A + m6 B + m6 C =
180◦ . 6 C is a right angle so m6 C = 90◦ . Therefore, m6 A + m6 B = 90◦ . Angles A and B are complementary angles
because their sum is 90◦ .
In Example A you saw that sin A = cos B and sin B = cos A. This means that the sine and cosine of complementary
angles are equal.
Example C
Find 80◦ and cos 10◦ . Explain the result.
Solution: sin 80◦ ≈ 0.985 and cos 10◦ ≈ 0.985. sin 80◦ = cos 10◦ because 80◦ and 10◦ are complementary angle
measures. sin 80◦ and cos 10◦ are the ratios of the same sides of a right triangle, as shown below.
Concept Problem Revisited
∆ABC is a right triangle with m6 C = 90◦ and sin A = k. What is cos B?
6
A and 6 B are complementary because they are the two non-right angles of a right triangle. This means that sin A =
cos B and sin B = cos A. If sin A = k, then cos B = k as well.
Vocabulary
The tangent (tan) of an angle within a right triangle is the ratio of the length of the side opposite the angle to the
length of the side adjacent to the angle.
The sine (sin) of an angle within a right triangle is the ratio of the length of the side opposite the angle to the length
of the hypotenuse.
The cosine (cos) of an angle within a right triangle is the ratio of the length of the side adjacent to the angle to the
length of the hypotenuse.
The trigonometric ratios are sine, cosine, and tangent.
Trigonometry is the study of triangles.
θ, or “theta”, is a Greek letter. In geometry, it is often used as a variable to represent an unknown angle measure.
Two angles are complementary if the sum of their measures is 90◦ .
Guided Practice
1. If sin 30◦ = 21 , cos? = 21 .
2. Consider the right triangle below. Find tan A and tan B.
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Chapter 7. Trigonometry
3. In general, what is the relationship between the tangents of complementary angles?
Answers:
1. The sine and cosine of complementary angles are equal. 90◦ − 30◦ = 60◦ is complementary to 30◦ . Therefore,
cos 60◦ = 21 .
2. tan A =
a
b
and tan B = ab .
3. In general, the tangents of complementary angles are reciprocals.
Practice
1. How are the two non-right angles in a right triangle related? Explain.
2. How are the sine and cosine of complementary angles related? Explain.
3. How are the tangents of complementary angles related? Explain.
Let A and B be the two non-right angles in a right triangle.
4. If tan A = 12 , what is tan B?
7
10 , what is cos B?
cos A = 14 what is sin B?
sin A = 35 , cos? = 35 ?
5. If sin A =
6. If
7. If
sin A+cos B
.
2
2
9. If tan A = 3 what is tan B?
10. If tan B = 15 , what is tan A?
8. Simplify
Which angle is bigger, 6 A or 6 B?
Solve for θ.
11. cos 30◦ = sin θ
12. sin 75◦ = cos θ
13. cos 52◦ = sin θ
14. sin 18◦ = cos θ
15. cos 49◦ = sin θ
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7.4. Inverse Trigonometric Ratios
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7.4 Inverse Trigonometric Ratios
Here you will learn how to use the inverses of the trigonometric ratios to find the measures of angles within right
triangles.
The maximum slope of a wheelchair ramp is 1:12. For a wheelchair ramp made with these specifications, what angle
does the ramp make with the flat ground?
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Click image to the left for more content.
http://www.youtube.com/watch?v=ypgcPKH4m4A James Sousa: Determine the Measure of an Angle of a Right
Triangle Using a Trig Equation
Guidance
Recall that the sine, cosine, and tangent of angles are ratios of pairs of sides in right triangles.
• The sine of an angle in a right triangle is the ratio of the side opposite the angle to the hypotenuse.
• The cosine of an angle in a right triangle is the ratio of the side adjacent to the angle to the hypotenuse.
• The tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side adjacent to the
angle.
You can use the trigonometric ratios to find missing sides of right triangles when given at least one side length and
one angle measure. You can use the inverse trigonometric ratios to find a missing angle in a right triangle when
given two sides.
• The inverse sine of a ratio gives the angle in a right triangle whose sine is the given ratio. Inverse sine is also
called arcsine and is labeled sin−1 or arcsin.
• The inverse cosine of a ratio gives the angle in a right triangle whose cosine is the given ratio. Inverse cosine
is also called arccosine and is labeled cos−1 or arccos.
• The inverse tangent of a ratio gives the angle in a right triangle whose tangent is the given ratio. Inverse
tangent is also called arctangent and is labeled tan−1 or arctan.
Note that in each case the “-1” is to indicate inverse, and is not an exponent.
To find the measure of an angle using an inverse trigonometric ratio, you will need to use your calculator. Most
scientific and graphing calculators have buttons that look like [sin−1 ], [cos−1 ], and [tan−1 ]. You will want to make
sure that your calculator is in degree mode so that the angle measure that the calculator produces is in degrees.
Example A
Solve for θ.
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