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Trigonometric Functions The Unit Circle The unit circle is a circle with radius one, centred at the point (0, 0). The equation of the unit circle is x2 + y 2 = 1 √ For example the point ( √ 3 6 , ) 3 3 is on the unit circle. Why? For every real positive number t, we can start from the point (1, 0) and move counter-clockwise around the unit circle for t units. The point we arrive at is called the terminal point of t. If t is negative, we travel clockwise. This process gives us an arc on the circle of length t. Recall that the circumference of the unit circle is 2π. Example: Find the terminal points for the following numbers. Draw the unit circle for each. arc 2π π π/2 3π/2 0 3π −π/2 −π terminal point 1 Example Find the terminal point for t = π/4. Similarly, we can prove the following table. arc terminal point 0 (1, 0) √ 3 1 π/6 (√2 ,√2 ) π/4 ( 22 ,√22 ) π/3 ( 12 , 23 ) π/2 (0, 1) Important Observation: For any real number t, the terminal point for t is equal to terminal point for t + 2kπ for any integer k. 2 Exercise Use the above table to find the terminal points of the following numbers. t = −π/4 t = −5π/6 t = 5π/6 t = 7π/4 t = 29π/6 3 Trigonometric Functions We learned that to every real number, we can assign a point (the terminal point) on the unit circle. We use the x and y coordinates of this point to define several functions. Let P (x, y) be the point on the unit circle defined by t. The trigonometric functions are defined as follows: 1. The function that assigns the value of y to t is called the sin function sin(t) = y 2. The function that assigns the value of x to t is called the cos function cos(t) = x 3. The function that assigns the value of y x tan(t) = y x 4. The function that assigns the value of x y cot(t) = x y 5. The function that assigns the value of 1 y csc(t) = 1 y 6. The function that assigns the value of 1 x sec(t) = 1 x to t is called the tan function to t is called the cot function to t is called the csc function to t is called the sec function Example: Find the value of all the trig functions for t = 4 π 3 and t = π. Observe that the functions sec and tan are not defined for those values of t whose terminal point has the x-coordinate 0. Also, the functions csc and cot are not defined for those values of t whose terminal point has the y-coordinate 0. Use the unit circle to show that the domains of the trig functions are as follows: Dtan Dsin = Dcos = R π = Dsec = {t | t 6= + kπ, k ∈ Z} 2 Dcot = Dcsc = {t | t 6= kπZ} 5 The table below gives you some of the values of trig functions. The sign of the trig functions at a given point t depends on the quadrant in which the terminal point of t lies in. To remember which function is positive in which quadrant, you can travel counter clockwise around the circle and say ”All Students Take Calculus“. Example: Determine the sign of the cos(t), sin(t) tan(t) and cot(t) at the following values for t. t = π4 , t= 2π , 3 t = − π6 , 5π . 6 6 In general, we have sin(−t) = − sin(t) csc(−t) = − csc(t) cos(−t) = cos(t) tan(−t) = − tan(t) sec(−t) = − sec(t) cot(−t) = − cot(t) Exercise: Use the unit circle to justify the above equalities. In general we have the following fundamental properties. csc t = 1 , sin t sec t = 1 , cos t cot t = 1 tan t sin t cos t , cot = cos t sin t 2 2 sin t + cos t = 1 tan t = Example: If cos t = functions at t. 3 5 and t is in the 4th quadrant, find all other trig 7