Chapter 4: Trigonometric Functions
1. When working many problems in trigonometry, it is really necessary to know the vocabulary of angles, . Some of the different types of angles are defined below.
Acute--any angle between 0 and 90 degrees
Right--a 90 degree angle
Obtuse--any angle between 90 and 180 degrees
Straight--a 180 degree angle
2. Two angles which have the same initial and terminal sides, but different rotations are called coterminal angles. It is always possible to find a coterminal angle for any given angle by either adding or subtracting .
3. Two positive angles are complements if their sum is .
Two positive angles are supplements if their sum is .
4. Another way to measure an angle is in radians. One radian is the measure of the central angle of a circle that intercepts an arc equal in length to the radius of the circle. If an arc of length s is measured on a circle of radius r, then the measure of the central angle that intercepts the arc is radians.
5. Many times it is necessary to convert from radians to degrees or degrees to radians. The conversion factors below can be used for these purposes.
1) To convert degrees to radians, multiply degrees by . (Degrees will cancel out, leaving radians.)
2) To convert radians to degrees, multiply radians by . (Radians will cancel out, leaving degrees.)
6. When an arc of length s is measured on a circle of radius r, the measure of the central angle that intercepts the arc is radians. The length of the arc intercepted by the central angle is .
7. If a point is in motion on a circle of radius r through an angle of radians in time t, then its linear speed is , where s is the arc length given by , and its angular speed is . The linear speed can also be written in terms of angular speed using , where is the angular speed in radians per unit of time.
||Q: Find a positive angle less than that is coterminal to .|
||Q: Find the complement and supplement, in that order, of .|
||Q: Convert into a radian measure in terms of .|
||Q: Convert into a degree measure.|
1. A unit circle is a circle of radius 1, with center at the origin of a rectangular coordinate system. The equation of this unit circle is . In a unit circle, the radian measure of a central angle is equal to the measure of its intercepted arc.
2. If t is a real number and P=(x, y) is a point on the unit circle that corresponds to t, then the trigonometric functions can be defined as follows.
3. The domain of the sine function and the cosine function is the set of all real numbers. The range of these functions is the set of all real numbers from -1 to 1, inclusive.
4. The cosine and secant are even functions. The sine, cosecant, tangent, and cotangent functions are odd.
5. The trigonometric functions are periodic functions. This means that they each have a pattern that is repeated again and again. The length of the pattern is called the period. The table below gives the period for each function.
6. There are many trigonometric identities. The fundamental identities, which are used quite frequently are listed below by category. It is very important that you know these fundamental identities.
It is also helpful to know the algebraic manipulations of some of these identities.
7. The repetitive behavior of the sine, cosine, and tangent functions are modeled by the equations in the box at the top of page 448.
8. When using a calculator to find the value of a trigonometric function, you must be sure that your calculator is in the correct mode, degrees or radians.
9. When using a calculator to find the value of the cotangent, cosecant, or secant of any angle, you must know which trig. functions are reciprocals of these three. Calculators do not have keys that will directly evaluate these three functions.
||Q: Use even and odd properties to find the exact value of the expression.|
||Q: Use periodic properties to find the exact value of .|
||Q: The height of the water H in feet at a boat dock t hours after 6 a.m. is given by . Find the height of the water at the dock at 10 a.m.|
||Q: Use an identity to find the exact value of the
1. In solving certain kinds of problems, sometimes it is
helpful to interpret the six trigonometric functions in right triangles
where angles are limited to acute angles. The inputs
for the functions are the measures of acute angles
in right triangles. The outputs are
the ratios of the lengths of the sides of right
triangles. The ratio of lengths depends on an angle and is a function
of the angle. If the angle is ,
then the six trigonometric functions are functions of . They are defined on page
454. It is extremely important that
you learn these definitions.
2. The trigonometric function values do not depend on the size of any triangle. They only depend on the size of the angle.
3. Within the six trigonometric functions, there are three sets of reciprocals. It is very helpful to know which functions are reciprocals of each other.
4. , , and are sometimes referred to as special angles. The values of their trig. functions can be found on page 457. They can also be derived using formulas for the trig. functions and the triangles on page 456. You need to know the exact value of the trig. functions for these special angles. You need to decide which method of learning them is best for you. You can memorize the functions for each angle, or you can memorize the triangles and derive them using the formulas.
5. If two angles are complements, the sine of one equals the cosine of the other. The secant of one equals the cosecant of the other. The tangent of one equals the cotangent of the other. Because of these relationships, trigonometric functions can be listed as cofunctions of each other, and the value of a trigonometric function of is equal to the cofunction of the complement of . The cofunction identities are given on page 458.
6. Many application problems use right triangle trigonometry. Some of these applications refer to an angle of elevation or an angle of depression.
Angle of elevation--the angle formed by a horizontal line and the line of sight to an object that is above the horizontal line.
Angle of depression--the angle formed by a horizontal line and the line of sight to an object that is below the horizontal line.
Q: Find a cofunction with the same value as the given
Q: A tower that is 152 ft tall casts a shadow 210 ft
long. Find the angle of elevation of the sun to the nearest tenth
of a degree.
1. In this section you will define the trigonometric functions of any angle. On page 465, the definitions of the trig. functions are restated. This time they are given in terms of a x, y, and r. (x, y) is a point on the terminal side of . and is the distance from (0, 0) to (x, y). Using these formulas, if you are given a point on the terminal side of , you can find the six trig. functions of . You should commit these formulas to memory, also.
2. Since r cannot be zero, the sine and cosine functions are defined for any angle. However, if the x-coordinate of the point on the terminal side of the angle is 0, then the tangent and secant functions are undefined. This will occur when is a quadrantal angle with its terminal side on the y-axis (Example: ). If the y-coordinate of the point is 0, then the cotangent and cosecant functions are undefined. This will occur when is a quadrantal angle with its terminal side on the x-axis (Example: )
3. It is very helpful when working certain problems to know if a trigonometric function is positive or negative. If is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which lies. Figure 4.34 on page 468 summarizes the signs of the trigonometric functions. Basically, though, if you know in which quadrants x and y are positive and negative, and you know the trig. functions formulas in terms of x and y, you can decide the sign of any trig. function without referring to Figure 4.34.
4. You can use an angle called a reference angle to find a trigonometric function for any angle . A reference angle is the positive acute angle formed by the terminal side of and the x-axis. Since you always find the reference angle from the x-axis, below are the rules for finding the reference angle in the second, third, and fourth quadrants. Reference angles are unnecessary for first quadrant angles.
5. When using a reference angle to find the value of a trig. function follow the steps below.
1. Find .
2. Find the trig. function value for .
3. Decide whether the trig. function will be positive or negative based on the quadrant that lies in.
||Q: (-4, 7) is a point on the terminal side of . Find the six trigonometric functions of .|
||Q: Find the exact value of , if and is in Quadrant III.|
||Q: Give the reference angle for and then evaluate .|
1. In order to graph the sine or cosine functions, it is important to first identify the amplitude and the period of the function. For the general forms given below, the formula for the amplitude of both the sine and cosine functions is . The formula for the period of both the sine and cosine functions is .
2. The steps for graphing variations of the sine function are on page 477. These steps require you to find five key points. For the sine function, three of these points will be x-intercepts. One will be the maximum point and one will be the minimum point. The same set of steps can be used for the cosine function. However, for the cosine only two points will be x-intercepts.
3. When finding your five key points it is important that you are able to work fractions. This is because in order to find the points, you must first take one-fourth of the period. Then, you must add this value to the x-value where the first cycle begins and keep adding it until you have all five x-values for the five key points.
Ex.: , for this function the period is . One-fourth of the period is . This value must be added to 0 first. It is then added again and again, as shown below, until you have five points in all.
After finding your x-values, you must substitute them into the function to find their corresponding y-values.
4. The formula for the phase shift of the sine or cosine functions is . The phase shift is always the beginning point in a function's period. It can be the first point or starting point of your five key points.
Ex.: , for this function the phase shift is . This could be used as your first x-value.
Remember! In the example above, C is not negative. In the general form, there is subtraction in the parentheses. Therefore, C is positive, which makes the function's phase shift to the right.
5. In the general form, D is the vertical shift of the function.
Ex.: , for this function D = -1. Therefore, the function's vertical shift is down 1.
||Q: Graph the function.|
||Q: Graph the function.|
6. The table on page 505 summarizes the graphs of the six trigonometric functions with a description of the domain, range, and period of the function.
||Q: Graph the function.
||Q: Graph the function.
||Q: Graph the function.
The same thing is true, if you are taking the inverse trig. function of same trig. function. See the example below.
||Q: Use a calculator to find the value of the expression to 2
||Q: Find the exact value of the expression
||Q: Use a sketch to find the exact value of the expression.
1. When you solve a right triangle, you find all its missing angles measurements and all the missing lengths of its sides.
2. When solving right triangles, it is very important that you know the formulas for the trig. functions in terms of side opposite, side adjacent and hypotenuse. These formulas are found on page 454.
3. Remember! Bearing is always measure from the north-south line either toward the east or toward the west. When writing bearing always give north or south first.
Ex.: is .
4. An object is in simple harmonic motion if its distance from the origin, d, at time t is given by either or . The frequency of the object is given by or .
||Q: Solve the right triangle, ABC, if C is the
right angle, c= 95.3, and .
||Q: An object moves in simple harmonic motion described by the
given equation, where t is measured in seconds and d in
inches. Find the maximum displacement, the frequency, and the time
required for one cycle (the period).
||Q: A boat leaves the entrance to a harbor and travels 210 miles on a bearing . How many miles north, and how many miles west from the harbor has the boat traveled?|