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MA 154 Section 6.3 Lesson 4 Delworth Trigonometric Functions of Real Numbers I want to talk about the basis of trigonometry, the unit circle. Draw a coordinate plane and only mark one unit in each direction.. Draw a circle that is centered at the origin and has a radius of one. This is the unit circle. Mark you x-axis and y-axis and also write cosine next to the x and sine next the y. Draw a right triangle with a central angle = 60, mark point P on the circle. The x-value of the coordinate is equal to the cos(60) and the y-value of the coordinate is equal to sin(60) Therefore the cos(60) = 3 1 and the sin(60) = 2 2 If we draw the corresponding right triangle for a 45 central angle, the corresponding point 1 1 1 1 is P , , therefore cos(45) = and sin(45) = 2 2 2 2 Draw the corresponding right triangle for a 30 central angle and the corresponding point is 3 1 3 1 P , , therefore cos(30) = 2 and the sin(30) = 2 2 2 3 sin 60 3 2 sin Since tangent = then; tan(60) = 2 3, 1 cos60 2 1 cos 2 1 1 sin( 45) sin 30 1 2 1 3 tan(30)= and, tan(45)= 2 1 2 1 cos(45) cos30 3 2 3 3 3 2 2 1 MA 154 Lesson 4 Section 6.3 Delworth Trigonometric Functions of Real Numbers Find the exact value of x and y. 8 y y y 5 x 60 30 x 45 x 6 y We can pick points on the unit circle that are not in QI and find the sine and cosine of the angle. Find the exact value of the six trigonometric functions: 3 = 90 = = 0 2 cos = 1 , tan < 0 2 csc = x 5 , sec < 0 2 Find the exact value of: cos(45) and cos(– 45) sin(45) and sin(– 45) tan(45) and tan(–45) Therefore: cos(–) = cos() sin(–) = – sin() tan() = – tan() 2 MA 154 Section 6.3 Lesson 4 Delworth Trigonometric Functions of Real Numbers A point P(x, y) is on the unit circle. Find the value of the six trig functions. 4 3 P , 5 5 8 15 P , 17 17 Let P(t) be a point on the unit circle. Find: P(t + ), P(t – ), P(–t), P(–t – ) 8 15 , 17 17 Let P be a point on the unit circle. Find the exact values of the six trig functions. 3 4 Use the formula for negative to find the exact value. cos135 sin 6 Verify the identity: csc(–x)cos(–x)=–cot(x) 3 3 2 tan 2 MA 154 Section 6.3 Lesson 4 Delworth Trigonometric Functions of Real Numbers y sine x cosine 4