1. 7 Trigonometric Identities and Equations
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2. Trigonometric Identities and Equations7
7.1 Fundamental Identities
7.2 Verifying Trigonometric Identities
7.3 Sum and Difference Identities
7.4 Double-Angle and Half-Angle Identities
7.5 Inverse Circular Functions
7.6 Trigonometric Equations
7.7 Equations Involving Inverse Trigonometric Functions
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3. Fundamental Identities
7.1
Fundamental Identities
Fundamental Identities ▪ Using the Fundamental Identities
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4. Fundamental Identities
Reciprocal Identities
Quotient Identities
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5. Fundamental Identities
Pythagorean Identities
Negative-Angle Identities
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6. Note
In trigonometric identities, θ can be an angle in degrees, an angle in radians, a real number, or a variable.
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7. If and θ is in quadrant II, find each function value.
Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT
If and θ is in quadrant II, find each function value.
(a) sec θ
Pythagorean identity
In quadrant II, sec θ is negative, so
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8. Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
(b) sin θ
Quotient identity
Reciprocal identity
from part (a)
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9. Example 1
FINDING TRIGONOMETRIC FUNCTION VALUES GIVEN ONE VALUE AND THE QUADRANT (continued)
(b) cot(– θ)
Reciprocal identity
Negative-angle identity
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10. Caution
To avoid a common error, when taking the square root, be sure to choose the sign based on the quadrant of θ and the function being evaluated.
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11. Express cos x in terms of tan x.
Example 2
EXPRESSING ONE FUNCTiON IN TERMS OF ANOTHER
Express cos x in terms of tan x.
Since sec x is related to both cos x and tan x by identities, start with
Take reciprocals.
Reciprocal identity
Take the square root of each side.
The sign depends on the quadrant of x. +
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12. Example 3
REWRITING AN EXPRESSION IN TERMS OF SINE AND COSINE
Write tan θ + cot θ in terms of sin θ and cos θ, and then simplify the expression.
Quotient identities
Write each fraction with the LCD.
Pythagorean identity
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13. For example, we would not write An argument such as θ is necessary.
Caution
When working with trigonometric expressions and identities, be sure to write the argument of the function.
For example, we would not write An argument such as θ is necessary.
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