Math 26: Fall (part 1) The Unit Circle: Cosine and Sine (Evaluating Cosine and Sine, and The Pythagorean Identity)

Transcription

1 Math : Fall 0 0. (part ) The Unit Circle: Cosine and Sine (Evaluating Cosine and Sine, and The Pthagorean Identit) Cosine and Sine Angle θ standard position, P denotes point where the terminal side of θ intersects the unit circle. -coordinate of P is the cosine of θ (Often written cos(θ) or cos θ) -coordinate of P is the sine of θ (Often written sin(θ) or sin θ) Different angles will give different and coordinates: (cos θ, sin θ) - sin(θ) cos(θ) θ - cos(θ) θ sin(θ) - - (cos θ, sin θ)

2 Eample: For each of the following angles: Draw α in standard position Label the coordinates of P (the point where the terminal side of α intersects the unit circle) Find sin(α) and cos(α). α =. α = Eample: Special Triangles:. For the iscosceles right triangle pictured below: a a (a) Use what ou know about triangles to determine the measure of angle (b) Use the pthagorean theorem to determine the length a.. Consider the equilateral triangle below: α α a α (a) What is the measure of α? (b) What is the measure of? (c) What is the length of side a? (d) Use the pthagorean theorem to find the length of the dashed blue line.

3 Using Special Triangles: to evaluate sin(θ) or cos(θ) of nice angles. Draw θ in standard position on the unit circle. Draw the point P (where the terminal side of θ intersects the unit circle) Draw a vertical line connecting P to the ais. Label the resulting triangle using one of the special triangles. Label the and coordinates of P. Answer the questions ou were asked (Find sin(θ) or cos(θ)) Triangle Triangle Evaluate sin( ) using the above method. - - ( cos( ), sin( )) - (, ) -

4 Eample: 5. Find the cosine and sine of the following angles: (a) α = (b) = (c) γ = (d) (optional) θ = (e) (optional) ρ = (f) (optional) σ = 5

5 The Reference Angle for an angle α is the angle is the Quadrant I angle that is the same as the acute angle between the terminal side of α and the ais. We don t need reference angles for non-quadrant Angles. Another wa to think about the reference angle for α is it s the Quadrant I angle that results from reflecting α over the -ais, -ais (or both). Based on the smmetr we see in the unit circle, reference angles help us evaluate cos or sin, but ma get the sign (±) wrong. Eample: Sketch the following angles in standard position, then find the Reference Angle for the angles below:. =. ρ = 5. (optional) θ = Finding Reference Angles: (Angles measured in radians) Reduced fraction multiples of with a denominator of have as a reference angle, those with a denominator of have as their reference angle, and those with a denominator of have as their reference angle. 5

6 Eample: Find sine and cosine of the following angles.. = 5. ρ = 5

7 The First of Man Trig Identities: Note, when ou square a number, ou lose the information about if it was positive of negative, so ou ll need to pa close attention to signs (±) when using the Pthagorean Identit. Eample:. Using the given information about θ, find the indicated value: (a) If θ is a Quadrant II angle with sin(θ) =, find cos θ. 5 (b) If < θ < and cos(θ) = 5, find sin(θ). 7 (c) If θ is a Quadrant IV angle with sin(θ) =, find cos(θ). 7