5.-The Sine and Cosine Functions Objectives:. Evaluate Sine and Cosine Functions. Find the sign of a Sine or Cosine function.. Use reference angles to evaluate sine or cosine function.. Evaluate expressions containing sine or cosine functions. 5. Use the fundamental identity to find the value of sine or cosine function. 6. Use trig functions to model the motion of a spring and other applications. Overview: In this set of notes we will define two trigonometric functions whose domain is the set of all angles. These two functions form the foundation of trigonometry. Definition: Sine and Cosine Functions: If α is an angle in standard position and (x, y) is the point of intersection of the terminal side and the unit circle, then. sine of α abbreviated sin(α) or sin α is the y-coordinate of that point.. cosine of α abbreviated cos(α) or cos α is the x-coordinate of that point. sin α = y, cos α = x. Domain and Range: The domain of the sine function and the cosine function is the set of angles in standard position, but since each angle has a measure in degrees or radians, we generally use the set of degree measures or the set of radian measures as the domain. If (x, y) is on the unit circle, then x and y, so the range of each of these functions is the interval [, ].
Example: Find the sine and cosine of the angle that passes through the point, circle. on the unit Solution: From the definition of sine and cosine: cosα = x cosα and sinα = sinα = y Sine and Cosine of Quadrantal Angles: Because the coordinates of the points where the unit circle crosses the x-axis and y-axis are easily determined, the sine and cosine of the quadrantal angles are also easy to determine from the unit circle. Example: Find the following trigonometric functions:. sine 9. cosine 8 π. cosine. sine π Solution: From the unit circle:. sine 9 =. cosine 8 = - π. cosine =. sine π = Notice that the sine and cosine of the quadrantal angles will always be ± or. When determining the sine or cosine of a quadrantal angle, use the following procedure:. Sketch the unit circle.. Label the coordinates where the unit circle crosses the axes.. Determine which axis the angles terminal side lies on.. Assign the appropriate value.
Example: Find the following trigonometric functions:. sine 5. sine 5. cosine. cosine 7 π Solution: Sketch a unit circle, label the coordinates, and then determine which of the axes the angle lies on. Assign the appropriate value.. sine 5. sine 5 =. cosine =. cosine 7 π = - Signs of Sine and Cosine: The signs of the sine and cosine functions depend on the quadrant in which the angle lies. The diagram below summarizes these sign values. Instead of memorizing the values in this table, it may be easier to remember the definitions of the functions. Because sin α = y, the sine function will have the same sign value as y in any quadrant. Likewise, because cos α = x, the cosine function will have the same sign value as x in any quadrant.
Sine and Cosine of 5 Degrees: Since the terminal side of a 5 degree angle lies on the line y = x, the x and y coordinates at the point of intersection with the unit circle are equal. The equation of the unit circle is x + y = and because y = x, we can use substitution to rewrite the equation as x + x = x x = x = ± x = ± = Because y = x, the coordinates of the point at the intersection of the unit circle and y = x is:,, And therefore sin 5 = and cos 5 = Sine and Cosine of a Multiple of 5 Degrees: There are four points where the lines y = x and y = -x intersect the unit circle. The coordinates of these key points can be determined as above and are shown on the diagram below. This diagram shows the coordinates of the key points for determining the exact value of the sine and cosine of any angle that is a multiple of 5 degrees or π radians. Example: Find the exact value of. 5 sin π.. 7 cos π Solution: From the diagram we can see that:.. sin cos =
Sine and Cosine of and 6 Degrees: Consider an equilateral triangle which has three sides of equal length (c = ) and all three angles are 6 degrees. Now cut this triangle in half, creating two right triangles with angles of, 6, and 9 degrees. It can be seen that the side opposite the degree angle (side a) will be half the length of side c. Using the Pythagorean Theorem, we can find the third side of the right triangle side b. a b b = + b = From this work, we can determine the coordinates of the point where a 6 degree angle intersects the unit circle to be P =, Therefore, sin 6 = and cos 6 = + b = c =
We can use the same -6-9 degree triangle to determine the point of intersection of the terminal side of a degree angle with the unit circle. From the diagram, the coordinates of the point where a degree angle intersects the unit circle are P =, Therefore, cos = and sin = Notice the relationship between the sine and cosine of and 6 degree angles. This is a significant relationship which should prove helpful for remembering these values. sin = cos6 and sin6 = cos Sine and Cosine of a Multiple of and 6 Degrees: The sine and cosine values of multiples of degree and 6 degree angles can be determined from the diagrams below. Please do not try to memorize these values. Once you understand the relationship demonstrated in these diagrams it will be easy to determine the function values of multiples of and 6 degree angles by simply sketching the angle. Multiples of Degrees Multiples of 6 Degrees
Example: Find the exact value of the following angles. 5 sin π. cos π 6 Solution: From the diagram we determine that: sin = 6 π cos Strategy for Finding Exact Values of Certain Angles: When determining the exact sine or cosine values of a, 5, or 6 degree angle, use the following procedure:. Determine if the angle is a multiple of, 5, or 6 degrees.. Sketch the angle in the appropriate quadrant on the unit circle.. Assign the appropriate value. Example: Find the sine and cosine of Solution: Because the denominator is a, this angle is a multiple of 6 degrees or π. The angle lies in Quadrant IV. Therefore sin and cos = Example: Find the sine and cosine of Solution: Because the denominator is a, this angle is a multiple of 5 degrees or π. The angle lies in Quadrant II. Therefore 5 sin π = and 5 cos π
Reference Angles: Another way to find the sine and cosine of an angle is to find the sine and cosine of the corresponding reference angle. A reference angle is the positive acute angle formed by the terminal side of an angle and the closest x-axis Definition: Reference Angle If θ is a nonquandrantal angle in standard position, then the reference angle for θ is the positive acute angle θ' (read theta prime ) formed by the terminal side of θ and the positive or negative x-axis. Theorem: Evaluating Trigonometric Functions Using Reference Angles For an angle θ in standard position that is not a quadrantal angle, the value of a trigonometric function of θ can be found by finding the value of the function for its reference angle θ' and prefixing the appropriate sign. Procedure for Finding Sine/Cosine Using Reference Angles:. Sketch the terminal side of the given angle in the appropriate quadrant.. Determine the angle measure between the terminal side and the nearest x-axis.. The sine/cosine of the given angle is the same as that of the reference angle after adjusting the sign value for that quadrant. Example: For each of the following, identify the quadrant the terminal side is in, find the reference angle and then determine the sine and cosine of the angle. 5 π 6 Solution: Sketch each angle in the appropriate quadrant than determine the positive angle measure between the angle and the nearest x-axis. This angle measure is the reference angle. II 8 = 6 sin = sin 6 = cos cos 6 III 8 = sin sin cos cos 5 IV 6 5 = 5 sin 5 sin 5 cos 5 = cos 5 = III π π = π sin sin π cos cos IV 6π π + = sin sin cos = cos = π 6 IV π π π = 6 6 6 π π sin sin 6 6 π π cos = cos = 6 6
Approximate Values for Sine and Cosine: The sine and cosine for any angle that is a multiple of, 5 or 6 degrees can be found exactly. These angles are so common that it is important to know these exact values. However, for most other angles we use approximate values for sine and cosine, found with the help of a scientific calculator. Example: Find each function value rounded to four decimal places. a. sin ( 8 ) b. cos ( - ) c. sin (.) d. cos (.5 ) Solution: Using a scientific calculator, we obtain: a. sin ( 8 ) =.978 b. cos ( - ) =.9 c. sin (.) =.6 d. cos (.5 ) =.8 The Fundamental Identity of Trigonometry: An identity is an equation that is satisfied for all values of the variable for which both sides are defined. The most fundamental identity in trigonometry involves the squares of the sine and cosine functions. The equation of the unit circle is x + y =. Because sin α = y and cos α = x we can write the following identity called the fundamental identity of trigonometry. sin α + cos α = This identity is mathematically useful in many ways. One such use is to find the value of sine or cosine if the value of the other function is known. Example: Find sin (α), given that cos (α) = 7 and α in quadrant IV. Solution: Use the fundamental identity of trigonometry to solve for sin α + cos α = sin sin sin α + = 7 6 α 9 α = 9 sinα = ± 7 sin α. Because the angle is in Quadrant IV, the sine is negative. Therefore, sinα 7
Modeling the Motion of a Spring: The sine and cosine functions are used in modeling the motion of a spring. If the weight is at rest while hanging from a spring then it is at an equilibrium position, or on a vertical number line. If the weight is set in motion with an initial velocity v from location x then the location at time t is given by v x = sin( ω t) + x cos( ω t) ω where ω is the spring constant, and t is time. A downward initial velocity is considered positive and an upward initial velocity is negative. Positive values of x are below the equilibrium position and negative values are above. Example: A weight on a vertical spring is given an initial downward velocity of 9 cm/sec from a point cm above equilibrium. Assuming that the spring constant has a value of ω =.5, write the formula for the location of the weight at time t. Find its location at seconds after it is set in motion. Write your answer using only exact values. Solution: Use the formula: Using v 9 and x =. = v x = sin( ω t) + x cos( ω t) ω x = 9 sin(.5t) cos(.5t).5 Solving the above equation for x at t=. v x = sin( ω t) + x cos( ω t) ω 9 x = sin(.5 ) cos(.5 ).5 x = sin 5 cos 5 x = x = x = 5 5 Therefore the spring will be located 5 cm. below the equilibrium point seconds after the spring is released.
Problem: A weight on a vertical spring is given an initial downward velocity of 7 cm/sec from a point cm below equilibrium. If the spring constant has a value of ω =, write the formula for the location of the weight at time t. Then, find its location 5 seconds after it is set in motion. Write your answer using only exact values. Solution: Use the formula: v x = sin( ω t) + x cos( ω t) ω where v is an initial velocity from location x, ω is the spring constant, and t is time. A downward initial velocity is considered positive and an upward initial velocity is negative. Positive values of x are below the equilibrium position and negative values are above. Using v = 7 and x = the formula is: 7 x = sin(t ) + cos(t ) Solving the above equation for t=5 seconds.. 7 x = sin( 5) + x = 6sin 6 + x = 6 x = x = + + cos 6 cos( 5) Therefore the spring will be located cm. below the equilibrium point seconds after the spring is released.
Example: The formula d = v sin( ) θ gives the distance (d) in feet that a projectile will travel when its launch angle is θ and its initial velocity is v feet per second. Approximately what initial velocity in feet per second does it take to throw a javelin 56 feet with launch angle 5 degrees? Solution: Let d = 56 and θ = 5 in the formula. d = v sin(θ ) 56 = v sin( 5) 56 = v sin() 56 = v 6,8 = v v = 8 The initial velocity is 8 feet per second.