**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.

**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.

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