Question

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

Accepted Answer

Identify the coordinates of the point where the terminal side of the angle $$ 	heta $$ intersects the unit circle. Since the circle has radius 1, these coordinates are $$ (x, y) $$ with $$ x = \cos(	heta) $$ and $$ y = \sin(	heta) . $$Recall the definitions of the six circular (trigonometric) functions in terms of $$ x $$ and $$ y  on $$the unit circle:

$$ \sin(	heta) = y $$
$$ \cos(	heta) = x $$
$$ 	an(	heta) = \frac{y}{x} \quad 	ext{(provided } x 
eq 0) $$
$$ \csc(	heta) = \frac{1}{\sin(	heta)} = \frac{1}{y} \quad 	ext{(provided } y 
eq 0) $$
$$ \sec(	heta) = \frac{1}{\cos(	heta)} = \frac{1}{x} \quad 	ext{(provided } x 
eq 0) $$
$$ \cot(	heta) = \frac{1}{	an(	heta)} = \frac{x}{y} \quad 	ext{(provided } y 
eq 0) $$ Substitute the $$ x $$ and $$ y $$ values from the point on the unit circle into each of the six function formulas to find their values. Check the quadrant in which the terminal side lies to determine the signs (positive or negative) of each function value, since sine and cosine can be positive or negative depending on the quadrant. Write down the six values: $$ \sin(	heta) $$, $$ \cos(	heta) $$, $$ 	an(	heta) $$, $$ \csc(	heta) $$, $$ \sec(	heta) $$, and $$ \cot(	heta) $$, ensuring to note any undefined values if division by zero occurs.

Each figure shows an angle θ in standard position with its terminal side intersecting the unit circle. Evaluate the six circular function values of θ.

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주요 개념

Unit Circle and Standard Position

Circular Functions (Trigonometric Functions)

Evaluating Trigonometric Functions from Coordinates