13.6 – 13.6 – The Tangent The Tangent FunctionFunction

I. I. Graphing the Tangent Graphing the Tangent FunctionFunction

The tangent of an angle is derived The tangent of an angle is derived from the coordinates of a point on a from the coordinates of a point on a line that is tangent to a circle.line that is tangent to a circle.

Tangent Tangent θθ – the y – coordinate of – the y – coordinate of the point where the line containing the point where the line containing the terminal side of the angle the terminal side of the angle intersects the tangent line x = 1.intersects the tangent line x = 1.

THINK: TANGENT = THINK: TANGENT = OPPOSITE/ADJACENTOPPOSITE/ADJACENT

Therefore, the tangent is equal to Therefore, the tangent is equal to

sin/cossin/cos

Sketch the graph of the tangent Sketch the graph of the tangent function, using a table from 0 function, using a table from 0 2 2ππ

Notice as you graph the tangent Notice as you graph the tangent function it, approaches infinity, and function it, approaches infinity, and thus has asymptotes at specified thus has asymptotes at specified values.values.

II. II. Properties of Properties of y = a tan b y = a tan b θθ

There is no amplitudeThere is no amplitude

Once cycle occurs in the interval from Once cycle occurs in the interval from

(-(-ππ/2b) to (/2b) to (ππ/b)/b)

ππ/b is the period of the function/b is the period of the function

there is vertical asymptotes at each end of there is vertical asymptotes at each end of the cyclethe cycle

Example 1: graph the following:Example 1: graph the following:

– A) y = 50 tan A) y = 50 tan θθ – B) f(x) = tan (B) f(x) = tan (ππ/6)/6)θθ

Example 2: solve the following:Example 2: solve the following:

A) tan A) tan θθ = 2 = 2B) 6 tan B) 6 tan θθ = 1 = 1