m12pc37

SECT. 3.7 TANGENT FUNCTION

REVIEW: TANGENT FUNCTION & TANGENT LINES

SOH-CAH-TOA

Using the ratio of opposite over adjacent, we can showthat the tangent function is also sine over cosine

A tangent line is a line that crosses a circle at onlyone point

Tangent Line

Point of Tangency

II) TANGENT FUNCTIONS AND UNIT CIRCLES

There are two ways to relate the tangent function with anunit circle

1st Method

The tangent function can be defined as the y-coordinate ofPoint P, divided by its x-coordinate

2nd Method

The tangent function can be defined as the y-coordinate ofthe extension of the terminal arm on a vertical tangent line

III) TANGENT = Y-COORDINATE / X-COORDINATE

The tangent function is the ratio of the Y-coordinateover the X-coordinate

As we rotate the around the unit circle, the ratiochanges according to the angle in standard position

IV) TANGENT AS THE Y-COORDINATE ON THE VERTICALTANGENT LINE

The tangent function can be defined as the y-coordinate of theextension of the terminal arm on a vertical tangent line

When you rotate the terminal arm,the extension creates a righttriangle with a base of 1

The tangent function will be theY-coordinate of this point on theVertical tangent

As the terminal arm rotates around the circle, it will cross thetangent line above the X-axis (+’ve) or below the X-axis (–’ve )

Ie: when the terminal arm is in Q1, it crosses the tangent lineabove the X-axis, so tanθ is positive

In Q2, the terminal armextends to the tangent linebeneath the X-axis, so itwill be negative

In Q3, it crosses thetangent line above the X-axis, so tanθ is positive

In Q4, it crosses thetangent line under the X-axis, so tan θ is negative

At any point when theterminal arm is pointingside ways, the value of tanθis equal to zero

At any point when it’spointing up or down, thevalue of tanθ is undefined,because it doesn’t touch thetangent line

TANGENT FUNCTION WITH SPECIAL TRIANGLES:

The Tangent of any angle with a reference angle of 30°,45°, or 60° involves the use of special triangles

Ex: Find the exact value of the following: (No Calculators)

Draw the angle in standard position

Find the reference angle

Draw the 30°, 45°, 60° triangle

Draw the angle in standard position

Find the reference angle

Draw the 30°, 60°, 90° triangle

Draw the angle in standard position

Find the reference angle

Draw the 45°, 45°, 90° triangle

Ex: Given that tan θ = 3/5 find all the possible coordinatesfor point P(x,y) in the unit circle

Determine which quadrant θ will be in

The ratiois positive

Find the coordinates using sine & cosine

The possible coordinates for P are:

Find the angles

HOMEWORK:

P 209 to 210

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