Lesson The Amazing Unit Circle: Trigonometric Identities
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<font size="+1"><strong>The unit circle definition of the trigonometric functions provides a lot of information</strong></font> <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/ucdefp.gif"></div> The trigonometric functions sine and cosine are defined in terms of the coordinates of points lying on the unit circle x<sup>2</sup> + y<sup>2</sup> = 1. Start by constructing the ray from the origin at angle θ (measured counter-clockwise from the positive x-axis). This ray meets the unit circle at a point P = (x,y). Cosine of the angle θ is <em>defined</em> to be the horizontal coordinate x of this point P: cos(θ) = x. Sine of the angle θ is <em>defined</em> to be the vertical coordinate y of P: sin(θ) = y. As θ changes so does the position of the point P and thus the values of cos(θ) = x and sin(θ) = y also change. In this way the <em>functions</em> f(&theta) = cos(θ) and g(&theta) = sin(θ) are defined. Right away the unit circle gives us properties of the cosine and sine functions. Since the point P lies on the unit circle, both the cosine and sine functions have <em>range</em> -1 to 1. We also see that the points at the right, top, left and bottom of the circle give the values: cos(0) = cos(0°) = 1 and sin(0) = sin(0°) = 0; cos(π/2) = cos(90°) = 0 and sin(π/2) = sin(90°) = 1; cos(π) = cos(180°) = -1 and sin(π) = sin(180°) = 0; cos(3π/2) = cos(270°) = 0 and sin(3π/2) = sin(270°) = -1. <strong>In this lesson we focus on trigonometric identities that can be immediately "read off" the unit circle.</strong> <strong>1. The Fundamental Trigonometric Identity</strong> Because the point P = (x,y) lies on the unit circle, these coordinates satisfy the equation x<sup>2</sup> + y<sup>2</sup> = 1. We have x = cos(θ) and y = sin(θ), and therefore <strong>cos<sup>2</sup>(θ) + sin<sup>2</sup>(θ) = 1.</strong> This is called <strong>the fundamental trigonometric identity.</strong> <strong>2. Complementary Angle Identities</strong> <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/uccomp3.gif"></div> Two angles are <em>complementary</em> if they add to make a right angle (π/2 = 90°). The angle complementary to θ is the angle π/2 - θ = 90° - &theta. To put the angle complementary to θ in standard position, start by reflecting the angle θ in the line y = x, which bisects the first quadrant. The angle BOQ is θ, so the angle AOQ measures π/2 - θ = 90° - θ. Thus Q has coordinates (cos(π/2-θ),sin((π/2-θ)) = (cos(90°-θ),sin((90°-θ)). When a point is reflected in the line y = x, its coordinates are reversed. So Q also has coordinates (sin(θ),cos(θ)). Therefore: <strong>cos(π/2-θ) = sin θ</strong> & <strong>sin(π/2-θ) = cos θ</strong> or <strong>cos(90°-θ) = sin θ</strong> & <strong>sin(90°-θ) = cos θ</strong>. These are the <strong>complementary angle identities</strong>. Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the complementary angle identities hold for all angles θ. <strong>3. Supplementary Angle Identities</strong> <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/ucsup3.gif"></div> Two angles are <em>supplementary</em> if they add to make a straight angle (π = 180°). The angle supplementary to θ is the angle π - θ = 180° - &theta. To put the angle supplementary to θ in standard position, start by reflecting the angle θ in the y-axis. The angle BOQ is θ, so the angle AOQ measures π - θ = 180° - θ. Thus Q has coordinates (cos(π-θ),sin(π-θ)) = (cos(180°-θ),sin(180°-θ)). When a point (a,b) is reflected in the y-axis, it moves to the point (-a,b). So Q also has coordinates (-cos(θ),sin(θ)). Therefore: <strong>cos(π-θ) = - cos θ</strong> & <strong>sin(π-θ) = sin θ</strong> or <strong>cos(180°-θ) = - cos θ</strong> & <strong>sin(180°-θ) = sin θ</strong>. These are the <strong>supplementary angle identities</strong>. Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the supplementary angle identities hold for all angles θ. <strong>4. Phase Shift Identities</strong> <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/ucperp3.gif"></div> The identities in this section are about angles that differ by a right angle. Start with the angle θ in standard position. Add a right angle to θ, so angle AOC measures θ + π/2 = θ + 90° and C has coordinates (cos(θ + π/2),sin(θ + π/2)) = (cos(θ + 90°),sin(θ + 90°)). Since angle AOB is a right angle, the angle BOC is &theta. Rotating the angle AOP by a right angle counterclockwise around the origin shows that the length OE = OD = cos(θ) and the length EC = DP = sin(θ). Thus C also has coordinates (-sin(θ),cos(&theta)). Therefore: <strong>cos(θ+π/2) = - sin θ</strong> & <strong>sin(θ+π/2) = cos θ</strong> or <strong>cos(θ+90°) = - sin θ</strong> & <strong>sin(θ+90°) = cos θ</strong>. Two more identities can now be obtained easily. <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/ucperp7.gif"></div> The angle AOF is obtained by <em>subtracting</em> a right angle from &theta, so F has coordinates (cos(&theta - π/2),cos(&theta - π/2)). Notice that &theta - π/2 = &theta - 90° and &theta + π/2 = &theta + 90° differ by a straight angle. This means the points C and F are on opposite sides of the unit circle, and so the coordinates of F have the opposite signs of the coordinates of the point C, that is, F also has coordinates (sin(θ),-cos(&theta)). Therefore: <strong>cos(θ-π/2) = sin θ</strong> & <strong>sin(θ-π/2) = - cos θ</strong> or <strong>cos(θ-90°) = sin θ</strong> & <strong>sin(θ-90°) = - cos θ</strong>. Although the diagram shows the angle θ in the first quadrant, the same conclusions can be reached when θ lies in any quadrant, and so these identities hold for all angles θ. The reason for calling these identities <strong>phase shift identities</strong> becomes clear when the functions of cos θ and sin &theta are graphed: the identities show that the graph of cos θ is the same as that of sin(&theta + π/2), which is the graph of sin &theta translated π/2 to the left. Such a translation is called a <em>phase shift</em>. Other identities, such as the opposite angle identities, can also be recognized as phase shift identities (with different phase shifts). <strong>5. Opposite Angle Identities</strong> <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/ucopp2.gif"></div> By <em>opposite angles</em> we mean two angles that differ by a straight angle, that is, by π = 180°. The angle opposite θ has measure θ + π = θ + 180°. Thus Q has coordinates<br />(cos(θ + π),sin(θ + π)) = (cos(θ + 180°),sin(θ + 180°)). The point on the opposite side of the unit circle from (a,b) is the point (-a,-b). So Q also has coordinates (-cos(θ),-sin(θ)). Therefore: <strong>cos(θ+π) = -cos θ</strong> & <strong>sin(θ+π) = -sin θ</strong> or <strong>cos(θ+180°) = -cos θ</strong> & <strong>sin(θ+180°) = -sin θ</strong>. These are the <strong>opposite angle identities</strong>. Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the opposite angle identities hold for all angles θ. <strong>6. Negative Angle Identities (Symmetry of Sine and Cosine)</strong> The <em>negative</em> -θ of an angle θ is the angle with the same magnitude but measured in the opposite direction from the positive x-axis. A positive angle θ is measured counterclockwise from the positive x-axis, so then -θ is measured clockwise from the positive x-axis. <div style="float:right;"><img src="http://mathmistakes.info/facts/TrigFacts/learn/images/ucsym2.gif"></div> The negative angle -θ is also the angle found by reflecting the angle θ in the x-axis. If the angle AOP is θ, then the angle AOQ is -θ. Thus Q has coordinates (cos(-θ),sin(-θ)). When a point (a,b) is reflected in the x-axis, it moves to the point (a,-b). So Q also has coordinates (cos(θ),-sin(θ)). Therefore: <strong>cos(-θ) = cos θ</strong> & <strong>sin(-θ) = - sin θ.</strong> These are the <strong>negative angle identities</strong>. Although the diagram shows the angle θ in the first quadrant, the same conclusion can be reached when θ lies in any quadrant, and so the negative angle identities hold for all angles θ. The negative angle identities also tell us the symmetries of the cosine and sine functions. A function f is <em>even</em> if f(-x) = f(x) for every x in the domain of f. Since cos(-θ) = cos θ, we conclude that <strong>cosine is an even function</strong>. An even function y = f(x) is symmetric about the y-axis, since for each point (x,f(x)) on the graph of f, the point (-x,f(-x)) = (-x,f(x)) also lies on the graph, so for each x the y-coordinate at -x is the same as the y-coordinate at x. A function f is <em>odd</em> if f(-x) = -f(x) for every x in the domain of f. Since sin(-θ) = - sin θ, we conclude that <strong>sine is an odd function</strong>. An odd function y = f(x) is symmetric about the origin, since for each point (x,f(x)) on the graph of f, the point (-x,f(-x)) = (-x,-f(x)) also lies on the graph, so for each x the y-coordinate on the graph at -x is the negative of the y-coordinate at x. The material in this lesson is adapted from <a href="http://mathmistakes.info/facts/TrigFacts/learn/uc/uc.html">The Amazing Unit Circle</a> at <a href="http://mathmistakes.info">mathmistakes.info</a>.
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