Trigonometric Functions The Unit Circle 4 2 Unit
- Slides: 18
Trigonometric Functions: The Unit Circle 4. 2
Unit Circle • The unit circle is a circle of radius 1 with its center at the origin.
Definitions of the Trigonometric Functions in Terms of a Unit Circle If t is a real number and P = (x, y) is a point on the unit circle that corresponds to t, then
Points on the Unit Circle
Text Example Use the Figure to find the values of the trigonometric functions at t= /2. Solution: The point P on the unit circle that Corresponds to t= /2 has coordinates (0, 1). We use x=0 and y=1 to find the Values of the trigonometric functions
The Domain and Range of the Sine and Cosine Functions • The domain of the sine function and the cosine function is the set of all real numbers • The range of these functions is the set of all real numbers from -1 to 1, inclusive.
Trigonometric Functions at /4.
Even and Odd Trigonometric Functions The cosine and secant functions are even. cos(-t) = cos t sec(-t) = sec t The sine, cosecant, tangent, and cotangent functions are odd. sin(-t) = -sin t csc(-t) = -csc t tan(-t) = -tan t cot(-t) = -cot t
Example • Use the value of the trigonometric function at t = /4 to find sin (- /4 ). Solution: sin(-t) = -sin t, so sin(- /4 ) = -sin( /4 ) = - 2/2
Reciprocal Identities
Quotient Identities
Example • If sin t = 2/5 and cos t = 21/5, find the remaining four trig functions
Pythagorean Identities
Example • Find cos t if sin t = 4/5
Definition of a Periodic Function A function f is periodic if there exists a positive number p such that f(t + p) = f(t) For all t in the domain of f. The smallest number p for which f is periodic is called the period of f.
Periodic Properties of the Sine and Cosine Functions sin(t + 2 ) = sin t and cos(t + 2 ) = cos t The sine and cosine functions are periodic functions and have period 2. sin = sin 3
Periodic Properties of the Tangent and Cotangent Functions tan(t + ) = tan t and cot(t + ) = cot t The tangent and cotangent functions are periodic functions and have period . tan = tan 2
Repetitive Behavior of the Sine, Cosine, and Tangent Functions For any integer n and real number t, sin(t + 2 n) = sin t, cos(t + 2 n ) = cos t, and tan(t + n) = tan t
- Unit circle radians
- Trigonometric functions unit circle approach
- Trigonometric functions unit circle approach
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- Cos90
- Three basic trigonometric functions
- Trig function transformations
- How to find period from sine graph
- 12-7 graphing trigonometric functions answers
- 12-1 trigonometric functions in right triangles
- Implicit derivatives
- 9-5 practice graphing trigonometric functions
- Domain and range of trigonometric functions examples
- Trig limits
- How is tangent used in real life
- 4-5 graphing other trigonometric functions
- Parts of trigonometric functions
- Domain and range of trigonometric functions
- Sin inverse x