In this explainer, we will learn how to solve trigonometric equations using the double-angle identity.
Trigonometric equations have several real-world applications in various fields, such as physics, engineering, architecture, robotics, music theory, and navigation, to name a few. In physics, they can be used in projectile motion, modeling the mechanics of electromagnetic waves, analyzing alternating and direct currents, and finding the trajectory of a mass around a massive body under the force of gravity.
Let’s begin by recalling the trigonometric functions, whose double-angle identities we will examine in this explainer. Consider a right triangle.
The trigonometric functions can be expressed in terms of the ratio of the sides of the triangle as
These functions satisfy the following trigonometric identity:
We note that these trigonometric ratios are defined for acute angles , and the trigonometric functions for all values of are defined on the unit circle using right triangle trigonometry.
Suppose that a point moves along the unit circle in the counterclockwise direction. At a particular position on the unit circle with angle , the sine function is defined as and the cosine function as , as shown in the diagram above. In other words, the trigonometric functions are defined using the coordinates of the point of intersection of the unit circle with the terminal side of in standard position.
The domain is the set of possible inputs and the range is the set of possible outputs, given its domain. For the trigonometric functions, these are given by the following.
| Domain | Range | |
|---|---|---|
Since the tangent function is defined as the ratio of the sine and cosine functions, it is undefined when , the denominator, is zero. In other words, the tangent function has to exclude values of where in order to be well defined. This is why the domain of the tangent function is , which just means we subtract the values of where from the set of real numbers in order to exclude this from the input.
The trigonometric functions are periodic, which means if we add an integer multiple of , in radians, or to the angle , the value of the function stays the same:
We can see these directly from the unit circle definition of the trigonometric functions. In fact, the tangent function is periodic by , in radians, or since we have
This fact will be important for finding general solutions for the trigonometric functions. The domains of the trigonometric functions have to be restricted to a particular subset, known as the principle branch, in order to have inverse functions.
The inverse trigonometric functions denoted by , , and are the inverse functions of the trigonometric functions , , and . This means they work in reverse or “go backward” from the usual trigonometric functions. They are defined by
These can also be written as , , and . The domain and range for the inverse trigonometric functions are given by the following.
| Domain | Range | |
|---|---|---|
The ranges for the inverse trigonometric functions in general apply only when the trigonometric functions are restricted to the principal branch. This is to ensure that the trigonometric functions are one-to-one functions, so that the inverse trigonometric functions evaluate to a single value, known as the principal value.
For example, if we have a particular trigonometric equation, such as we can find the solutions in the range by applying the inverse trigonometric equations:
However, if we want to determine all the possible solutions, we need the general solutions given in terms of an integer , which we can obtain from the CAST diagram and the periodicity of the trigonometric functions.
Let’s recall the CAST diagram.
Definition: The CAST Diagram
- In the first quadrant, all trigonometric functions are positive.
- In the second quadrant, the sine function is positive.
- In the third quadrant, the tangent function is positive.
- In the fourth quadrant, the cosine function is positive.
Let’s recall how we can find the solutions to trigonometric equations.
Property: Solutions to Trigonometric Equations
The CAST diagram helps us to remember the signs of the trigonometric functions for each quadrant.
In particular, the CAST diagram tells us that solutions to the trigonometric equations are given by the following.
- If and , for , or, in radians, for .
- If and , then we can express the angle in terms of the inverse cosine function in degrees as for , or, in radians, for .
- If , then we can express the angle in terms of the inverse tangent function in degrees as for , or, in radians, for .
The ranges given for follow from the ranges of the inverse trigonometric functions.
We can also see this from the unit circle as shown.
The general solutions to the trigonometric equations can be found from the solutions we obtain from the CAST diagram or inverse trigonometric functions, , by adding a integer multiple of or , in radians. We do this for all the solutions we obtain, since the trigonometric functions are periodic. Thus, the general solution, , for , is in degrees and in radians.
When solving trigonometric equations, we are usually given a particular range for the angle to determine the solutions, which means we may only need to consider a few values of , where appropriate. A solution set is the set of values that contains solutions to the trigonometric equation in the required range.
Now, let’s recall the addition identities for the sine, cosine, and tangent functions:
We will use these addition identities to derive the double-angle identities.
By substituting in the addition identities, we obtain the double-angle identities for the trigonometric functions.
Definition: Trigonometric Double-Angle Identities
The trigonometric double-angle identities are
Let’s consider an example that demonstrates how we can use the sine double-angle identity to solve a particular trigonometric equation in a specified range.
Example 1: Solving an Equation in a Specified Range Using the Double-Angle Identities
Find the set of possible solutions of given .
Answer
In this example, we are going to solve a trigonometric equation in a particular range using the double-angle identities.
The double-angle formula for sine is given by
Therefore, is equivalent to
The general solution, for (using the CAST diagram), to this equation can be found as or which is equivalent to or for . The first expression gives us and the second expression , for . For other integers , we would obtain angles outside the required range.
Thus, given , the possible solutions are
The double-angle identity can also be used to solve trigonometric equations of the form that we saw earlier, by squaring both sides and using the Pythagorean identity. Now, let’s consider an example where we demonstrate this to find the solutions of a trigonometric equation in this form.
Example 2: Solving Trigonometric Equations Involving Special Angles
If and , find the value of .
Answer
In this example, we are going to solve a trigonometric equation in a particular range using the double-angle identities.
In order to solve , we begin by squaring both sides and distributing:
On applying the Pythagorean identity and the double-angle identity , we have
The general solution, for (using the CAST diagram), to this equation can be found as or which is equivalent to or for . The second expression gives , for . For other integers , we would obtain angles outside the required range.
Thus, given , the only possible solution is
Let’s see how we can use the double-angle identities to solve other trigonometric equations in a particular range. As an example, suppose we want to find all the solutions within the range to the trigonometric equation
On applying the double-angle formula for cosine and using the Pythagorean identity, this can be written as
If we let , this is equivalent to solving the quadratic equation
We can solve this using the quadratic formula or factorization to give
Thus, the solutions are and . We can ignore the second solution since for , we have . Thus, we only consider the solutions with or for . The acute solution is given by
General solutions can be found using the CAST diagram and the periodicity of the sine function, for , as and
Now, we can substitute particular integer values of in order to find all the solutions within the required range. In particular, for and , we obtain the solutions and from the first and second expressions for the general solution respectively. For other integers , we would obtain angles outside the range .
To summarize, the solutions to , in degrees, for , are
Now, let’s look at a few more examples in order to practice and deepen our understanding of solving trigonometric equations using the double-angle identities.
In the next example, we will use the double-angle identity for sine to find the solutions, in degrees.
Example 3: Solving a Trigonometric Equation Using Double-Angle Identities
Find the set of solutions in the range for the equation .
Answer
In this example, we are going to solve a trigonometric equation in a particular range using the double-angle identities.
On distributing the parentheses on the left-hand side of the given trigonometric equation and applying the Pythagorean identity ,
The double-angle identity for sine is given by
On substituting this, we have and thus the given trigonometric equation is equivalent to
If we let , we have
The solutions are and . For , we have which has the general solution, for (using the CAST diagram), and
These two expressions are equivalent. For , we have which has the general solution and
To summarize, the general solutions, for , are
For , we obtain the solutions and , from the first and third expression respectively, and for , we obtain from the second solution. For other integers , we would obtain angles outside the required range.
Thus, given , the solutions are
Now, let’s look at an example where we will use the double-angle identity for cosine to find the solutions, in degrees, to a trigonometric equation. This time we also have to consider a quadratic equation and the range for the cosine function.
Example 4: Using Double-Angle Identities to Solve a Trigonometric Equation
Find the solution set for given , where .
Answer
In this example, we are going to solve a trigonometric equation in a particular range using the double-angle identities.
The double-angle formula for cosine is given by
On applying the Pythagorean identity, we can rewrite this as
On applying this to the left-hand side of the given trigonometric equation, we obtain
Thus, the given trigonometric equation can be rewritten as
If we let , this can be written as a quadratic equation:
We can find the solution to this quadratic equation using the quadratic formula to obtain
This gives and , but since and we have , then we can ignore the first solution and have to solve
The general solution, for (using the CAST diagram), can be written as and
The first expression gives and the second expression , for . For other integers , we would obtain angles outside the required range.
Thus, given , the possible solutions are
In the next example, we will see how we can use either the double-angle sine or cosine identity to solve a trigonometric equation, as it can be expressed in terms of both after some manipulation.
Example 5: Solving a Trigonometric Equation Using Double-Angle Identities
Find the set of possible values of that satisfy , where .
Answer
In this example, we are going to solve a trigonometric equation in a particular range using the double-angle identities.
Recall that the double-angle identity for sine is
Now, to solve the given trigonometric equation, we note that, using the Pythagorean identity , the denominator on the left-hand side can be written as
Note that the absolute value is necessary to account for the fact that the value of could be negative in the interval . Thus, using the double-angle identity, the left-hand side of the given trigonometric equation becomes
So, equating this to the right-hand side, we have
Let us consider the two possible values of case by case. Firstly, for , the general solution, for , is
Ordinarily, we would check the supplementary angle too, but since , this would result in an equivalent expression. Thus, the only two solutions to in the range are and (found by setting and 1 respectively). Now, let us consider . The general solution of this, for , is
And for the supplementary angle, we have
On closer inspection, we can see that these expressions are equivalent since . Thus, the two solutions to in the interval are and (found by setting and 1 respectively).
Combining the solutions for and together, we have
Some trigonometric equations may require the use of the half-angle identities for the trigonometric functions, which follow directly from the double-angle identities.
The half-angle identities for the trigonometric functions are given by the following.
Definition: Trigonometric Half-Angle Identities
The trigonometric half-angle identities are
These can be shown from the double-angle identities. For example, if we consider the double-angle identity for cosine, we rearrange this to make the subject of the equation:
Now, if we let , this can be written as which is the half-angle identity for sine. The other half-angle identities can be found in a similar way.
Now, let’s look at an example where we use the half-angle identity for cosine to solve a trigonometric equation, in radians.
Example 6: Solving Trigonometric Equations Involving Half Angles
By using the half angle formula , or otherwise, solve the equation , where .
Answer
In this example, we will solve a trigonometric equation in a particular range using the half-angle identity for cosine.
If we substitute the half-angle formula, the equation can be rewritten as
On squaring both sides, we have
If we let , we have to solve
The solutions are and . For , we have which has the general solution, for (using the CAST diagram), and
We note that the second expression is equivalent to the first; the general solution is an integer multiple of . For , we have which has the general solution, for , and
To summarize, the general solutions, for , are
For , we obtain the solutions , , and from the first, second, and third expressions respectively. For other integers , we would obtain angles outside the required range.
Thus, given , the solutions are
Finally, we will look at an example where we use the half-angle identity for the tangent function to solve a trigonometric equation, in radians.
Example 7: Solving Trigonometric Equations Involving Half Angles
Solve , where .
Answer
In this example, we are going to solve a trigonometric equation in a particular range using the half-angle identities.
The half-angle identity for the tangent function is given by
Thus, we have to solve
On applying the Pythagorean identity, we have
If we let ,
The solutions to this quadratic equation are and . For , we have which has the general solution, for (using the CAST diagram), and
For , we have which has the general solution and
We note that the second expression is equivalent to the first, that the general solution is an integer multiple of . To summarize, the general solutions, for , are
For , we obtain the solutions , , and from the first, second, and third expressions respectively. For other integers , we would obtain angles outside the required range.
Thus, given , the solutions are
Let us finish by recapping a few important key points from this explainer.
Key Points
- We can solve trigonometric equations using the double-angle or half-angle identities.
- The trigonometric double-angle identities are
- The trigonometric half-angle identities are These follow directly from the double-angle identities.
- After finding the principal value solution, in degrees or radians, we can find the general solution
for the trigonometric functions, for , using the CAST diagram and the periodicity of the trigonometric functions.
- We are usually given a particular range for the angle to determine the solutions, which means we only consider particular integers to find the possible solutions.
