Introduction
Some trigonometric equations cannot be readily solved by factoring. As an alternative method, the trigonometric equation should be rewritten in terms of one function. This can be done by substituting an existing expression with an equivalent identity. The object is to express the equation with only one function and then to apply the necessary skills of algebra to solve for that function and then use an inverse trigonometric function to solve for the variable.
Example 1:
Solve for all values of .
Solution: The equation now has two functions – sine and cosine. Study the equation carefully and decide in which function to rewrite the equation. is readily expressed in terms of cosine by using the Pythagorean Identity .
where is any integer.
Example 2:
Solve for
Solution: The equation now has two functions – cosine and tangent. Study the equation carefully and decide in which function to rewrite the equation. In this case we actually don’t need to change all of the functions to one, as the function can be separated by factoring. If the common factor were factored out, then a double angle identity for cosine could be substituted into the new expression.
Lesson Summary
In this lesson you learned that by substituting trigonometric identities into an equation provided you with one that could be solved. Without these substitutions, the trigonometric equations would be impossible to solve. You must be careful when doing using these identities to ensure that you make the correct substitution and use the applicable identity to achieve success.
Points to Consider
Is using the quadratic formula an option when solving a trigonometric equation?
Review Questions
Solve for all values of
Hint: and
Solve for all values of .
Hint: Use the double angle identity for
Solve the trigonometric equation over the interval .
Review Answers
Solving Trigonometric Equations (Using the Quadratic Formula)
Learning objectives
A student will be able to:
Solve trigonometric equations by using the quadratic formula.
Introduction
When solving quadratic equations that do not factor, the quadratic formula is often used. The same can be applied when solving trigonometric equations that do not factor. The values for is the numerical coefficient of the term, is the numerical coefficient of the function term and is a constant. The formula will result in two answers and both will have to be evaluated within the designated interval.
Example 1: Solve for exact values of over the interval .
Solution:
The equation will not factor. Use the quadratic formula for .
The tangent function is positive in the first and third quadrant.
Therefore
The tangent function is negative in the second and fourth quadrant.
Therefore
Example 2: Solve for values of over the interval
Solution:
There are no solutions for since is not in the range of values for
The sine function is positive in the first and second quadrants.
Therefore
Lesson Summary
In this lesson you have learned how to solve trigonometric equations that are quadratic. The same rules from algebra are used when the quadratic formula is used to solve a trigonometric function. Two solutions are obtained and these solutions must be adapted to the designated interval of the problem.
Points to Consider
Are there other methods that can be used to solve trigonometric equations?
Can these methods be applied to solve trigonometric equations that have multiple angles?
Review Questions
Solve for values of over the interval
Hint: Replace with
Solve for values of over the interval
Solve the trigonometric equation such that over the interval .
Review Answers
and
Hint: Replace with
and
Applications and Technological Tools
Learning objectives
A student will be able to:
Use technology to solve trigonometric equations
Example 1
Solve the equation over the interval
Solution:
Pythagorean Identity
Let Therefore:
Using the quadratic formula:
In radians, for And in the third quadrant
for And in the fourth quadrant
We can verify the solution graphically:
The solution agrees with the values of for which the graph of crosses the axis in the above graphs. The values pictured in the four smaller graphs were obtained by using the zoom feature of the TI-83.
Often, solving a trigonometric equation algebraically can be very involved and complicated. To solve the equation takes a great deal of skill and time. As an alternative to this long process, the equations can be readily solved by using technology. The trigonometric equation can be solved algebraically as well as by using technology. The following graph of the cubic function was created by using the software program – Autograph.
However, using a graphing calculator will produce the same graph. The intercepts can be determined by using the functions available on the calculator.
We will begin by solving the equation algebraically.
Solution:
The period of is which means that the solutions will repeat every .
Therefore the solutions are:
1.
2.
3.
4.
5.
6.
where is any integer.
Solving the equation algebraically was quite involved and required a lot of time to complete. Now, we will use the graphing calculator to solve the equations. The solutions will be estimates of the solutions.
The solutions are very close to those that resulted from the algebraic solution of the equation.
Lesson 6
Solve equations (with double angles)
Learning objectives
A student will be able to:
Use the double angle identities for the sine, cosine and tangent functions to solve trigonometric equations
Introduction
The double angle formulas can be used to compute exact values or to change the form of existing trigonometric equations. You learned about these formulas in the previous chapter, but we will briefly review them here before investigating how they will be useful when working solving equations involving trigonometric functions. These formulas are quite simple to derive , so we will start by deriving them again below, followed by examples which show how they can be used to solve equations.
Double Angle Identity for the Sine Function
One of the formulas for calculating the sum of two angles is:
If and are both the same angle in the above formula, then
This is the double angle formula for the sine function.
Example 1:
Find all solutions to the equation in the interval
Solution:
Apply the double angle formula
The values for in the interval are and
The values for in the interval are and
Double Angle Identity for the Cosine Function
Another formula for calculating the sum of two angles is:
If and are both the same angle in the above formula, then
This is one of the double angle formulas for the cosine function. Two more formulas can be derived by using the Pythagorean Identity
and likewise
The double angle formulas for are:
Example 2:
Find .
Solution:
Double Angle Identity for the Tangent Function
Another formula for calculating the sum of two angles is:
If and are both the
same angle in the above formula, then
Example 3:
If and is an acute angle, find the exact value of .
Solution: Cotangent and tangent are reciprocal functions. .
Therefore
Example 4:
Solve the trigonometric equation such that
Solution:
Lesson Summary
In this lesson we reviewed how to derive the double angle formulas (also referred to as the double angle identities) for the sine, cosine and tangent functions. Then we used these formulas to determine exact values, to solve equations and to write expressions. The more you use these formulas, the more adept you will become at manipulating them and at choosing the correct one to arrive at the solution for the problem.
Points to Consider
Are there similar formulas that can be derived for other angles?
How can these other formulas be used?
Review Questions
If and , use the double angle formulas to determine each of the following.
Use the double angle formulas to prove that the following equations are identities. Prove that the left hand side is equal to the right hand side by working with the left hand side only.
Solve the trigonometric equation such that
Solve the trigonometric equation such that
Review Answers
Hint: Replace with
and
Hint: Replace with
Vocabulary
Double Angle Identity the formulas that result from and being equal in the angle sum formulas.
Identity A statement of equality between two expressions that is true for all values of the variable for which the expressions are defined.
Solving Trigonometric Equations Using Half Angle Formulas
Learning objectives
A student will be able to:
Apply the half angle identities for the sine, cosine and tangent functions
Introduction
As you learned in the last chapter, the half angle formulas can be used to compute exact values or to simplify trigonometric expressions. As you remember, these formulas are quite simple to derive by using the double angle formulas and performing some manipulations. We will review these derivations and then apply the formulas to solve trigonometric equations.
Half-Angle Identity for the Sine Function
In the previous lesson, one of the formulas that was derived for the cosine of a double angle is:
if is located in either the first or second quadrant.
if is located in the third or fourth quadrant.
Example 1:
Use the half angle formula for the sine function to determine the value of .
Solution:
and the angle is located in the first quadrant. Therefore,
Another way to do this problem would be to use which is the value of the sine of as one of the special angles. The result would be . If this were entered into a calculator, the result would be the same as the first solution.
This value can also be determined by using a calculator but it is necessary to practice working with the formula.
Half-Angle Identity for the Cosine Function
In the previous lesson, one of the formulas that was derived for the cosine of a double angle is:
if is located in either the first or fourth quadrant.
if is located in either the second or fourth quadrant.
Example 2:
Use the half angle formula for the cosine function to prove that the following expression is an identity.
Solution:
Use the formula and substitute it on the left-hand side of the expression.
Half-Angle Identity for the Tangent Function
The half angle identity for the tangent function begins with the reciprocal identity for tangent.
The half angle formulas for sine and cosine are substituted into the identity.
There are two half angle formulas for the tangent function.
Example 3:
Without the use of technology, use the half-angle identity for tangent to determine an exact value for .
Solution:
Example 4:
Solve the trigonometric equation over the interval .
Solution:
or or These are the three solutions in and the following solutions are the result of the function being periodic.
Lesson summary
In this lesson you have learned how to derive the half-angle angle formulas (also referred to as the half- angle identities) for the sine, cosine and tangent functions. These formulas are used to determine exact values, to solve equations and to write expressions to prove that they are equal. Once again, practice makes perfect, so you will have to use these formulas in order to arrive at the correct solution for the various problems.
Points to Consider
All of the examples in both this lesson and the previous lesson dealt with single angles. Can these formulas be used to solve trigonometric equations when multiple angles are in the solution?
Review Questions
Without the use of technology, use the half-angle identities for the trigonometric functions to determine an exact value for each of the following:
Prove that
Solve the trigonometric equation such that
Review Answers
Solving Trigonometric Equations with Multiple Angles
Learning objectives
A student will be able to:
Solve equations with multiple angles by applying the half angle identities and the double angle identities for the sine, cosine and tangent functions
Introduction
The double angle and the half-angle identities can be used to compute exact values or to change the form of existing trigonometric equations. These formulas have been derived in the previous lessons and will be applied to problems in this lesson to demonstrate that they can work with other trigonometric formulas.
Example 1:
Find the exact value of cos given and is in quadrant .
Solution:
Example 2:
Solve the trigonometric equation over the interval .
Solution:
The solutions for are
The solutions for are
Lesson summary
In this lesson you have learned how to solve trigonometric equations with multiple angles. The methods used to solve these equations will be often used when solving trigonometric equations. The solutions that you present require a that you understand the defined interval for the values of the angle.
Points to Consider
Can technology be used to either solve these trigonometric equations or to confirm the solutions?
Review Questions
Solve the trigonometric equation over the interval .
Solve the trigonometric equation over the interval .
Solve the trigonometric equation for all real values of .
Solve the trigonometric equation such that
Review Answers
and
Hint: Rewrite the equation in terms of tan by using the Pythagorean identity
and where is any integer.
Hint: Use the double angle identity for .
and
Applications and Technological Tools
Learning objectives
A student will be able to:
Use technology to solve trigonometric equations.
Explore real life problems that involve solving trigonometric equations.
Example 1:
1. The range of a small rocket that is launched with an initial velocity at an angle with the horizontal is given by . If the rocket is launched with an initial velocity of , what angle is needed to reach a range of ?
Solution:
or
2. Using the TI-83 to solve a trigonometric equations is sometimes easier than solving the equation algebraically.
Solve such that using technology.
i.
ii.
/> iii. Use CALC to find the intersection points of the graphs.
3. Show that
Solution:
This can be verified graphically:
The graph of is the same as the graph of
4. A spring is being moved up and down. Attached to the end of the spring is an object that undergoes a vertical displacement. The displacement is given by the equation . Find the first two values of (in seconds) for which .
CK-12 Trigonometry Page 18
