Unit 7

Units 1 to 6 Review

Read

Part 7.1B corresponds to section 4.1 to 4.4 on page 166; section 5.1 on page 222; section 5.4 on page 266; and sections 6.1 to 6.4 on page 290 of your Pre-Calculus 12 textbook.

Angles and Angle Measure

Recall angles can be measured in degrees or in radians.

To convert from degrees to radians, multiply by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo»§#960;«/mo»«msup»«mn»180«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mfrac»«/mstyle»«/math». To convert from radians to degrees, multiply by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«msup»«mn»180«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«mo»§#960;«/mo»«/mfrac»«/mstyle»«/math».

Central angles and arc lengths are related by the equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»a«/mi»«mo»=«/mo»«mo»§#952;«/mo»«mi»r«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»a«/mi»«/mstyle»«/math» is the arc length, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»§#952;«/mo»«/mstyle»«/math» is the central angle in radians, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»r«/mi»«/mstyle»«/math» is the radius of the circle.

Angles of rotation start on the positive «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis. If the rotation is up (counter clockwise), it is a positive rotation. If the rotation is down (clockwise), it is a negative rotation.

The terminal arm is the stopping place of the angle of rotation. If two angles undergo a different number of rotations, but land at the same place, they are coterminal.

A principal angle is the shortest positive rotation from the initial arm (positive «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis) to the terminal arm.

A reference angle is the shortest angle measure from the positive or negative «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis to the terminal arm. It can be above or below the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-axis, but the reference angle is always positive and always less than «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»90«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mstyle»«/math».

The reference angles of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»30«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»45«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»60«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«/mstyle»«/math», or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfrac»«mo»§#960;«/mo»«mn»6«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mfrac»«mo»§#960;«/mo»«mn»4«/mn»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mspace width=¨0.33em¨/»«mi»and«/mi»«mspace width=¨0.33em¨/»«mfrac»«mo»§#960;«/mo»«mn»3«/mn»«/mfrac»«/mstyle»«/math» radians, are special angles that produce primary trigonometric ratios that can readily be determined and used as exact values.

Trigonometric ratios can be defined in terms of the intersection of the terminal arm of an angle in standard position and the unit circle, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mi»x«/mi»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»y«/mi»«/mrow»«/mfenced»«/mstyle»«/math». The unit circle can also be used to determine exact value ratios of special angles.

The primary trigonometric ratios are sine, cosine, and tangent.


Note the angles of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»0«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«mfenced»«mn»0«/mn»«/mfenced»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»90«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«mfenced»«mfrac»«mo»§#960;«/mo»«mn»2«/mn»«/mfrac»«/mfenced»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«msup»«mn»180«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«mfenced»«mo»§#960;«/mo»«/mfenced»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»270«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«mfenced»«mfrac»«mrow»«mn»3«/mn»«mo»§#960;«/mo»«/mrow»«mn»2«/mn»«/mfrac»«/mfenced»«/mrow»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mn»360«/mn»«mo accent=¨true¨»§#176;«/mo»«/msup»«mfenced»«mrow»«mn»2«/mn»«mo»§#960;«/mo»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» also have readily available trigonometric ratios associated with them.

The reciprocal ratios are cosecant, secant, and cotangent.


The reciprocal trigonometric ratios can be replaced with primary trigonometric ratios in equations that can be solved algebraically and graphically.