3.3 Definition III: Circular Functions

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# 3.3 Definition III: Circular Functions - PowerPoint PPT Presentation

3.3 Definition III: Circular Functions. A unit circle has its center at the origin and a radius of 1 unit. Circular Functions. Unit Circle. Domains of the Circular Functions. Assume that n is any integer and s is a real number. Sine and Cosine Functions: ( , )

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Presentation Transcript
3.3 Definition III: Circular Functions
• A unit circle has its center at the origin and a radius of 1 unit.
Domains of the Circular Functions
• Assume that n is any integer and s is a real number.
• Sine and Cosine Functions: (, )
• Tangent and Secant Functions:
• Cotangent and Cosecant Functions:
Evaluating a Circular Function
• Circular function values of real numbers are obtained in the same manner as trigonometric function values of angles measured in radians.
• This applies both to methods of finding exact values (such as reference angle analysis) and to calculator approximations. Calculators must be in radian mode when finding circular function values.
Example: Finding Exact Circular Function Values
• Find the exact values of
• Evaluating a circular function at the real number is equivalent to evaluating it at radians. An angle of intersects the unit circle at the point .
• Since sin s = y, cos s = x, and
Example: Approximating
• Find a calculator approximation to four decimal places for each circular function. (Make sure the calculator is in radian mode.)
• a) cos 2.01  .4252 b) cos .6207  .8135
• For the cotangent, secant, and cosecant functions values, we must use the appropriate reciprocal functions.
• c) cot 1.2071
The length s of the arc intercepted on a circle of radius r by a central angle of measure radians is given by the product of the radius and the radian measure of the angle, or s = r, in radians.

3.4 Arc Length and Area of a Sector

A circle has radius 18.2 cm. Find the length of the arc intercepted by a central angle having each of the following measures.

a)

b) 144

Example: Finding Arc Length
A rope is being wound around a drum with radius .8725 ft. How much rope will be wound around the drum it the drum is rotated through an angle of 39.72?

Example: Finding a Length
Two gears are adjusted so that the smaller gear drives the larger one, as shown. If the smaller gear rotates through 225, through how many degrees will the larger gear rotate?Example: Finding an Angle Measure
Solution
• Find the radian measure of the angle and then find the arc length on the smaller gear that determines the motion of the larger gear.
Solution continued
• An arc with this length on the larger gear corresponds to an angle measure , in radians where
• Convert back to degrees.
Area of a Sector
• A sector of a circle is a portion of the interior of a circle intercepted by a central angle. “A piece of pie.”
• The area of a sector of a circle of radius r and central angle  is given by
Example: Area
• Find the area of a sector with radius 12.7 cm and angle  = 74.