1. 1
Trigonometric Functions
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2. Trigonometric Functions1
1.1 Angles
1.2 Angle Relationships and Similar Triangles
1.3 Trigonometric Functions
1.4 Using the Definitions of the Trigonometric Functions
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3. Using the Definitions of the Trigonometric Functions
1.4
Using the Definitions of the Trigonometric Functions
Reciprocal Identities ▪ Signs and Ranges of Function Values ▪ Pythagorean Identities ▪ Quotient Identities
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4. Reciprocal Identities
For all angles θ for which both functions are defined,
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5. Since cos θ is the reciprocal of sec θ,
Example 1(a)
USING THE RECIPROCAL IDENTITIES
Since cos θ is the reciprocal of sec θ,
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6. Since sin θ is the reciprocal of csc θ,
Example 1(b)
USING THE RECIPROCAL IDENTITIES
Since sin θ is the reciprocal of csc θ,
Rationalize the denominator.
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7. Signs of Function Values
 in Quadrant
sin 
cos 
tan 
cot 
sec 
csc  I + + + + + + +  II  +
III  + IV
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8. Ranges of Function Values
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9. Example 2
DETERMINING SIGNS OF FUNCTIONS OF NONQUADRANTAL ANGLES
Determine the signs of the trigonometric functions of an angle in standard position with the given measure.
(a) 87°
The angle lies in quadrant I, so all of its trigonometric function values are positive.
(b) 300°
The angle lies in quadrant IV, so the cosine and secant are positive, while the sine, cosecant, tangent, and cotangent are negative.
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10. Example 2
DETERMINING SIGNS OF FUNCTIONS OF NONQUADRANTAL ANGLES (cont.)
Determine the signs of the trigonometric functions of an angle in standard position with the given measure.
(c) –200°
The angle lies in quadrant II, so the sine and cosecant are positive, while the cosine, secant, tangent, and cotangent are negative.
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11. Example 3
IDENTIFYING THE QUADRANT OF AN ANGLE
Identify the quadrant (or quadrants) of any angle  that satisfies the given conditions.
(a) sin  > 0, tan  < 0.
Since sin  > 0 in quadrants I and II, and tan  < 0 in quadrants II and IV, both conditions are met only in quadrant II.
(b) cos  > 0, sec  < 0
The cosine and secant functions are both negative in quadrants II and III, so  could be in either of these two quadrants.
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12. Ranges of Trigonometric Functions
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13. Decide whether each statement is possible or impossible.
Example 4
DECIDING WHETHER A VALUE IS IN THE RANGE OF A TRIGONOMETRIC FUNCTION
Decide whether each statement is possible or impossible.
(a) sin θ = 2.5
Impossible
(b) tan θ =
Possible
(c) sec θ = .6
Impossible
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14. Suppose that angle  is in quadrant II and
Example 5
FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
Suppose that angle  is in quadrant II and
Find the values of the other five trigonometric functions.
Choose any point on the terminal side of angle .
Let r = 3. Then y = 2.
Since  is in quadrant II,
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15. Remember to rationalize the denominator.
Example 5
FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Remember to rationalize the denominator.
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16. Example 5
FINDING ALL FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
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17. Pythagorean Identities
For all angles θ for which the function values are defined,
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18. Quotient Identities
For all angles θ for which the denominators are not zero,
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19. Example 6
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
Choose the positive square root since sin θ >0.
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20. To find tan θ, use the quotient identity
Example 6
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
To find tan θ, use the quotient identity
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21. Caution
In problems like those in Examples 5 and 6, be careful to choose the correct sign when square roots are taken.
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22. Find sin θ and cos θ, given that and θ is in quadrant III.
Example 7
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
Find sin θ and cos θ, given that and θ is in quadrant III.
Since θ is in quadrant III, both sin θ and cos θ are negative.
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23. Example 7
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Caution
It is incorrect to say that sin θ = –4 and cos θ = –3, since both sin θ and cos θ must be in the interval [–1, 1].
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24. Example 7
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Use the identity to find sec θ. Then use the reciprocal identity to find cos θ.
Choose the negative square root since sec θ <0 for θ in quadrant III.
Secant and cosine are reciprocals.
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25. Example 7
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
Choose the negative square root since sin θ <0 for θ in quadrant III.
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26. Example 7
FINDING OTHER FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
This example can also be worked by drawing θ is standard position in quadrant III, finding r to be 5, and then using the definitions of sin θ and cos θ in terms of x, y, and r.
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