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Trigonometric Functions 1. Definition of radians and conversion between radians and degrees. a. A radian= Measure of the angle when the arc subtended equals the radius of the circle. Hence, the length of the arc subtended by the angle divided by the radius of ArcLength the circle gives the measure of the angle in radians. It is Radius 0 2 radians 360 b. radians 180 0 0 radians 0 0 , 2. Common angles in radians 2 6 30 0 , 90 0 , 180 0 , 4 45 0 , 3 60 0 3 270 0 , 2 360 0 2 3. Unit circle. Definition of sine and cosine and their graphs. a. Use the coordinates and the triangle determined to show sine and cosine. Indicate the variation of sine and find the relationship with its representation in the Cartesian plane. Show the meaning of a period. Make sure the relation between what happens in the circle and the plane is understood including the signs of both functions. b. Fundamental trigonometric identity. sin 2 x cos 2 x 1 sin( ) sin( ) c. Other relations cos( ) cos( ) 4. Graphs of sine and cosine. a. Basic relation between sine and cosine (analysis through translations) sin( x 2 ) sin( x) sin( x cos( x 2 ) cos( x) 2 ) sin( x) b. New graphs under transformations. Period and amplitude. Determine the period and amplitude of the following functions: y sin( 3x ) 1 y 0.5 cos( x) 20 4 Trigonometric Functions 4/29/2017 Page1 4. Find two equations for the following graph. One in terms of sine and the other in terms of cosine 2 sin( 4 x ) 12 11 10 10 9 8 0.79 1.18 1.57 x 5. Table with important angles: 0 sin x 0 cos x 1 tan x 0 6 1 2 3 2 3 3 4 2 2 2 2 1 3 2 3 2 2 3 2 1 2 1 0 -1 0 0 -1 0 1 Und. 0 Und. 0 3 1.96 2.36 6. Reference angle of an angle in standard position. It is the smallest angle it makes with the x-axis. If is the reference angle for , sin( ) sin( ) except probably for the sign. II I β=α Sin (β) = Sin (α) Cos (β) = Cos (α) β α Sin (β) = Sin (α) Cos (β) = - Cos (α) IV III β α Sin (β) = -Sin (α) Cos (β) = -Cos (α) Trigonometric Functions 4/29/2017 β α Sin (β) = -Sin (α) Cos (β) = Cos (α) Page2 The reference angle is always between zero and / 2 . The diagram above shows the relationship between the value of the angle and its reference angle on each quadrant. For each of the angles given below o o o Draw each angle Indicate its reference angle Determine the value of the function using the reference angle. 2 5 10 sin( ) cos( ) sin( ) sin( ) 3 6 4 3 o Find all solutions for each of the equations: o sin( x) 1 o tan( x ) 1 1 o cos( x) 2 7. Definition of the other trig functions using triangles. Values for the common angles. Signs in each quadrant using ASTC. opp. hip. csc( ) Opp. hip. opp adj hip cos( ) sec( ) θ hip. adj opp. adj Adj. tan( ) cot( ) adj. opp. Using these definitions we have the following relationships among trigonometric functions: Sin ( x) 1 1 Cos( x) 1 Tan( x) Csc( x) Sec( x) Cot ( x) Cos( x) Sin ( x) Cos( x) Sin ( x) Tan( x) sin( ) Hip. II I Students Only sine (and csc are positive) All Are positive III Taking Only tan (and cot are positive) Trigonometric Functions 4/29/2017 IV Now we can use the reference angles to find the value of all the trigonometric functions. To help us doing this, we will remember the following rule ASTC (All Students Take Calculus) Calculus Only cosine (and sec are positive) Page3 EXERCISE: Find the value of the following trigonometric function, by first finding the reference angle: Tan( ) 5 ) 3 8 Cot ( ) 6 22 Sec ( ) 4 EXERCISE: In each case find an expression for each of the other trigonometric functions of all the trigonometric functions given the following information: o tan x 3x o tan 2 o sin 2x 2x o cos 3 Csc ( 8. Other important identities. tan 2 x 1 sec 2 x 1 cot 2 x csc 2 x sin( x y ) sin x cos y sin y cos x cos( x y ) cos x cos y sin x sin y sin( 2 x) 2 sin x cos x cos( 2 x) cos 2 ( x) sin 2 ( x) 1 2 sin 2 ( x) 2 cos 2 ( x) 1 1 cos( 2 x) sin 2 ( x) 2 1 cos( 2 x) cos 2 ( x) 2 EXERCISE: Use the identities above to simplify the given equations. Then solve them: o 2 cos 2 x 3 sin 2 x 3 o sin 2 x cos 2 x o cos 2 3 cos 2 9. Graphs of all the trig functions Trigonometric Functions 4/29/2017 Page4 10. Law of cosine 11. Law of sine 12. Inverse functions. Use definition to calculate some values. Trigonometric Functions 4/29/2017 Page5