3 Quadrant II Quadrant I 360º Quadrant III Quadrant IV Y-axis 90º r r θ Terminal siderrθ0ºX-axis180ºInitial side360ºQuadrant IIIQuadrant IV270º
4 Quadrant II Quadrant I Quadrant III Quadrant IV Y-axis -270º -360º X-axis-180ºTerminalsideInitial side0ºrθQuadrant IIIQuadrant IV-90º
5 angle θ…measured from positive x-axis, or initial side, to terminal sidecounterclockwise: positive directionclockwise: negative directionfour quadrants…numbered I, II, III, IV counterclockwise
6 six trigonometric functions for angle θ whose terminal side passes thru point(x, y) on circle of radius rsin θ = y / r csc θ = r / ycos θ = x / r sec θ = r / xtan θ = y / x cot θ = x / yThese apply to any angle in any quadrant.
7 For any angle in any quadrant x2 + y2 = r2 …So, r is positive by Pythagorean theorem.(x,y)ryθx
8 NOTE: right-triangle definitions are special case of circular functionswhen θ is inquadrant IY(x,y)ryθXx
9 *Reciprocal Identities sin θ = y / r and csc θ = r / ycos θ = x / r and sec θ = r / xtan θ = y / x and cot θ = x / y
10 *Both sets of identities are useful to determine trigonometric *Ratio Identities*Both sets of identities are usefulto determine trigonometricfunctions of any angle.
11 Students Take Classes Positive trig values in each quadrant: All YStudentsAllall six positivesin positive(csc)(-, +)(+, +)IIIXIIIIVTakeClasses(-, -)(+, -)tan positive(cot)cos positive(sec)
12 In the ordered pair (x, y), x represents cosine and REMEMBER:In the ordered pair (x, y),x represents cosine andy represents sine.Y(-, +)(+, +)IIIXIIIIV(-, -)(+, -)
14 #1 Draw each angle whose terminal side passes through the given point, and findall trigonometric functions of each angle.θ1: (4, 3)θ2: (- 4, 3)θ3: (- 4, -3)θ4: (4, -3)SOLUTION
15 x = y = I r = (4,3) θ1 sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = SOLUTION
16 x = II y = r = (-4,3) θ2 sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =SOLUTION
17 x = y = r = θ3 (-4,-3) III sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =θ3(-4,-3)IIISOLUTION
18 x = y = r = θ4 (4,-3) IV sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =θ4(4,-3)IVSOLUTION
19 Perpendicular II I line from point on circle always drawn to the x-axisforming areference triangleIIIref θ2θ1Xref θ3ref θ4IIIIV
20 is equal to trig function of its reference angle, or it differs Value of trigfunctionof angle inany quadrantis equal to trig function of itsreference angle,or it differsonly in sign.YIIIref θ2θ1Xref θ3ref θ4IIIIV
21 #2 Given: tan θ = -1 and cos θ is positive: Draw θ. Show the values for x, y, and r.SOLUTION
22 Given: tan θ = -1 and cos θ is positive: Find the six trigonometric functions of θ.SOLUTION
Your consent to our cookies if you continue to use this website.