5.3 Trigonometric Functions of Any Angle
How do you find the Values of the Trigonometric Functions for a Quadrantal Angle? Remember a quadrantal angle has its terminal side on the x-axis or y-axis.
1. Draw the angle in standard position. 2. Choose a point P on the angle's terminal side. 3. Choose a point that is 1 unit from the origin. 3. Apply the definitions of the appropriate trigonometric functions.
What are the Definitions of Trigonometric Functions of Any Angle?
Because the point P = (x,y) is any point on the terminal side of θ other than the origin, (0,0), r = [square root of x^2 + y^2]. Note that the denominator of the sine and cosine functions is r. Because r ≠ 0, the sine and cosine functions are defined for any angle θ. This is not true for the other four trigonometric functions. Note that the denominator in the tangent and secant functions is x: tan θ = y/x and sec θ = r/x. These functions are not defined if x = 0. If the point P = (x,y) is on the y-axis, then x= 0. Thus, the tangent and secant functions are undefined for all quadrantal angles with terminal sides on the positive or negative y-axis. Likewise, if P = (x,y) is on the x-axis, then y = 0, and the cotangent and cosecant functions are undefined: cot θ = x/y and csc θ = r/y. The cotangent and cosecant functions are undefined for all quadrantal angles with terminal sides on the positive or negative x-axis.
Example Evaluating Trigonometric Functions Using the Quadrant in Which θ or an Angle Lies
Given tan θ = -2/3 and cos θ > 0, find cos θ and csc θ. SOLUTION: Because the tangent is negative and cosine is positive, θ lies in quadrant IV. This helps determine whether the negative sign in tan θ = -2/3 should be associated with the numerator or the denominator. Keep in mind that in quadrant IV, x is positive and y is negative. Thus, tan θ = -2/3 = y/x = (-2)/3. *In quadrant IV, y is negative* SEE THE IMAGE FOR THE REST OF THE SOLUTION In the above example, we used the quadrant in which θ lies to determine whether a negative sign should be associated with the numerator or the denominator. Here's a similar situation, where negative signs should be associated with both the numerator and the denominator: tan θ = 3/5 and cos θ < 0. Because the tangent is positive and cosine is negative, θ lies in quadrant III. In quadrant III, x is negative and y is negative. Thus, tan θ = 3/5 = y/x = (-3)/(-5) *We see that x = -5 and y = -3*
What are the Signs of the Trigonometric Functions?
If θ is not a quadrantal angle (terminal side on the x or y-axis) 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. For example, if θ lies in quadrant II, x is negative and y is positive. Thus, the only positive ratios in this quadrant are y/r and its reciprocal, r/y. These ratios are the function values for the sine and cosecant, respectively. In short if θ lies in quadrant II, sin θ and csc θ are positive. The other four trigonometric functions are negative. Figure 5.38 summarizes the signs of the trigonometric functions. If θ lies in quadrant I, all six functions are positive. If θ lies in quadrant II, only sin θ and csc θ are positive. If θ lies in quadrant III, only tan θ and cot θ are positive. Finally, if θ lies in quadrant IV, only cos θ and sec θ are positive. Observe that the positive functions in each quadrant occur in reciprocal pairs.
Angles formed by P= (x,y), a point r units from the origin on the terminal side of θ.
Remember an angle is in standard position when it's initial side lies on the x-axis. The point P = (x,y) is a point r units from the origin on the terminal side of θ. A right triangle is formed by drawing a line segment from P = (x,y) perpendicular to the x-axis. Note that y is the length of the side opposite θ and x is the length of the side adjacent to θ. The definitions of the six trigonometric functions can be extended to include such angles as those in the image (and quadrantal angles, not pictured). Remember a quadrantal angle has its terminal side on the x-axis or y-axis. The point P= (x,y) may be any point on the terminal side of the angle θ other than the origin, (0,0).
How do you Evaluate 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 angle, θ', is always positive. However, the same function value for θ may be positive or negative. For example, we can use a reference angle, θ', to obtain an EXCACT value for tan 120°. The reference angle for θ = 120° is θ' = 180° - 120° = 60°. We know the exact value of the tangent function of the reference angle: tan 60° = (the square root of 3-learned in 5.2 using an equilateral triangle and the Pythagorean theorem.) We also know that the value of a trigonometric function of a given angle, θ, is the same as that of its reference angle, θ', except possibly for the sign. Thus, we can conclude that tan 120° equals the negative square root of 3 or the positive square root of 3. What sign should we attach to the square root of 3? A 120° angle lies in quadrant II, where only the sine and cosecant are positive. Thus, the tangent function is negative for a 120° angle. Therefore, tan 120° = -tan 60° = (the negative square root of 3) * Prefix by a negative sign to show tangent is negative to quadrant II. The reference angle for 120° is 60°. *Remember that we use two right triangles from a drawn equilateral triangle and the pythagorean theorem (a^2 + b^2 = c^2) to find exact trigonometric values of 30°, 45°, and 60°. For the exact value of 30° and 60° angles, use 2 for the hypotenuse (c) and 1 for the leg to find the other leg, which is the square root of 3. For the exact value of a 45° angle, use 1 for each leg (a and b) and the the hypotenuse (c) will be the square root of 2.
What is a Reference Angle and how do you find it?
We will often evaluate trigonometric functions of positive angles greater than 90° and all negative angles by making use of a positive acute angle. This positive acute angle is called a reference angle. Definition of a Reference Angle Let θ be a nonacute 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. Figure 5.40 shows the reference angle for θ lying in quadrants II, III, and IV. *Notice that the formula used to find θ', the reference angle, varies according to the quadrants in which θ lies.* You may find it easier to find the reference angle for a given angle, by making a figure that shows the angle in standard position. The acute angle formed by the terminal side of this angle and the x-axis is the reference angle.
