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# Trigonometric Identities Help (page 3)

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

## Trigonometric Identities Practice Problems

#### Practice 1

Illustrate, using the unit circle model, examples of the following facts:

sin – θ = –sin θ

cos – θ = cos θ

#### Solution 1

See Fig. 2-4. This shows an example for an angle θ of approximately 60° (π/3 rad). Note that the angle –θ is represented by rotation to the same extent as, but in the opposite direction from, the angle θ . Generally, positive angles are represented by counterclockwise rotation from the x axis, and negative angles are represented by clockwise rotation from the x axis. The ray from the origin for – θ looks like the reflection of the ray for θ from a pane of glass that contains the x axis and is perpendicular to the page. The above identities can be inferred geometrically from this diagram. The two rays intersect the circle at points whose y values (representing sines) are negatives of each other, and whose x values (representing cosines) are the same.

Fig. 2-4. Illustration for Solution 1.

#### Practice 2

Simplify the expression sin (120° – θ ). Express coefficients to three decimal places.

#### Solution 2

Use the formula for the sine of an angular difference, given above, substituting 120° for θ in the formula, and θ for ø in the formula:

In case you don’t already know this definition, a coefficient is a number by which a variable or function is multiplied. In the answer to this problem, the coefficients are 0.866 and 0.500.

#### Practice 3

Illustrate, using the unit circle model, examples of the following facts:

sin(180° – θ ) = sin θ

cos(180° – θ ) = –cos θ

#### Solution 3

See Fig. 2-5. This shows an example for an angle θ of approximately 30° ( π /6 rad). The ray from the origin for 180° – θ looks like the reflection of the ray for θ from a pane of glass that contains the y axis and is perpendicular to the page. The above identities can be inferred geometrically from this diagram. The two rays intersect the circle at points whose y values (representing sines) are the same, and whose x values (representing cosines) are negatives of each other.

Fig. 2-5 . Illustration for Solution 3.

Practice problems for these concepts can be found at:  Trigonometric Functions Practice Test

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