Algebra and Trig. I 4.4 Trigonometric Functions of Any Angle In the last section we looked at trigonometric functions of acute angles. Note the angles below are in standard position. IN this section we will be looking at angles that are not acute, however still in standard position. We can extend our definition of the six trigonometric equations to include such angles as well as quadrantal angles (such angles are angles that have the terminal side that lies on the y-axis or the x-axis) Definition of Trigonometric Functions of Any Angle Let θ be any angle in standard position and let be a point on the terminal side of θ. If is the distance from (0,0) to (x,y), the six trigonometric functions of θ are defined by the following ratios: Notice that the ratios in the second column are the reciprocals of the ratios in the first column. Note: r is any point other than (0,0) so therefore r 0. 1 P a g e
Example Let be a point on the terminal side of θ. Find each of the six trigonometric functions of θ. Example Let be a point on the terminal side of θ. Find each of the six trigonometric functions of θ. 2 P a g e
How to find the values of trigonometric functions at quadrantal angles? Step 1: Draw the angle in standard position Step 2: Choose a point that lies on the angle s terminal side. Because the trig. functions depend on θ and not on the distance of the point P from the origin, perhaps use the point that is one unit away from the origin. (i.e ) Step 3: Apply the definitions of the appropriate trigonometric functions. Example Evaluate, if possible, the sine function and the tangent function at the following four quadrantal angles. (use ) 1. 2. 3. 4. 3 P a g e
The Signs of the Trigonometric Functions If θ is not a quadrantal angle then the sign of a trigonometric function depends on the quadrant in which θ lies. In all four quadrants r is positive, however x and y can be positive or negative. Recall: QI positive x and positive y (+,+) QII negative x and positive y (-,+) QIII negative x and negative y (-,-) QIV positive x and negative y (+,-) So if we think of a point in QII, (-,+) the only trig. functions that are positive are sine and cosecant all others are negative. All trig. functions are positive in QI Sine and its reciprocal, cosecant are positive in QII Tangent and its reciprocal, cotangent are positive in QIII Cosine and its reciprocal, secant are positive in QIV Example If and, name the quadrant in which θ lies. 4 P a g e
Example If and, name the quadrant in which θ lies. Example Given and, find and Example Given and, find and 5 P a g e
Definition of a Reference Angle Let θ be a non-negative acute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle θ formed by the terminal side of θ and the x-axis. Example Find the reference angle θ, for each the following angles. a) c) b) d) 6 P a g e
Finding Reference Angles for Angles Greater than 360 (2π) or less than -360 (-2π) 1. Find a positive angle α less than 360 or 2π that is coterminal with the given angle. 2. Draw α in standard position 3. Use the drawing to find the reference angle for the given angle. The positive acute formed by the terminal side of α and the x-axis is the reference angle. Example Find the reference angle for each of the following angles a) c) b) d) 7 P a g e
Evaluating Trigonometric Functions Using Reference Angles The values of the trigonometric functions of a given angle, θ, are the same as the values of the trigonometric functions of the reference angle, θ, except possibly for the sign. A function value of the acute reference, θ, is always positive. However, the same function value for θ may be positive or negative. For example we can use a reference angle to obtain an exact value for tan120. The reference angle for θ=120 is θ =180-120 =60. We know the exact value of the tangent function of the reference angle: We also know that the value of a trig. function of a given angle, θ, is the same as that of its reference angle, θ, except possibly the sign. Thus we can conclude that So what sign should we attach to? A 120 angle lies in QII, where only the sine and cosecant are positive. Thus the tangent function is negative for a 120 angle. Therefore Procedure for Using Reference Angles to Evaluate Trigonometric Functions The value of a trigonometric function of any angle θ is found as follows: 1. Find the associated reference angle, θ, and the function value for θ 2. Use the quadrant in θ lies to prefix the appropriate sign to the function in step 1. 8 P a g e
Example Use reference angles to find the exact value of each of the following trigonometric functions: a) c) b) d) 9 P a g e
e) 10 P a g e