Solving trigonometric equations

Solving Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions. These equations can often be solved using algebraic methods and the properties of trigonometric functions. In this section, we will explore how to solve various types of trigonometric equations.

Basic Trigonometric Equations

The most common trigonometric equations involve the basic functions: sine (sin), cosine (cos), and tangent (tan). The general form of a basic trigonometric equation is:

sin(θ) = k, cos(θ) = k, or tan(θ) = k,

where k is a constant. The solutions to these equations are angles θ for which the sine, cosine, or tangent value equals k.

Solving sin(θ) = k

To solve the equation sin(θ) = k, follow these steps:

1. Determine the value of k. Remember that the range of sin(θ) is between -1 and 1. If k is outside this range, there are no solutions.
2. Find the reference angle. The reference angle is the angle whose sine value is |k|.
3. Identify the quadrants where sin(θ) is positive or negative.
4. Write the general solutions.

**Example 1:** Solve sin(θ) = 0.5.

1. Here, k = 0.5, which is within the range.

2. The reference angle is θ_ref = 30° (since sin(30°) = 0.5).

3. Sin is positive in the first and second quadrants.

4. The general solutions are:

θ = 30° + 360°n and θ = 180° - 30° + 360°n, where n is any integer.

Thus, θ = 30° + 360°n and θ = 150° + 360°n.

Solving cos(θ) = k

To solve the equation cos(θ) = k, use similar steps:

1. Check if k is within the range of -1 to 1.
2. Find the reference angle.
3. Identify the quadrants where cos(θ) is positive or negative.
4. Write the general solutions.

**Example 2:** Solve cos(θ) = 0.5.

1. Here, k = 0.5, which is within the range.

2. The reference angle is θ_ref = 60° (since cos(60°) = 0.5).

3. Cos is positive in the first and fourth quadrants.

4. The general solutions are:

θ = 60° + 360°n and θ = 360° - 60° + 360°n.

Thus, θ = 60° + 360°n and θ = 300° + 360°n.

Solving tan(θ) = k

To solve tan(θ) = k, follow these steps:

1. Check the value of k, as tan can take any real number.
2. Find the reference angle.
3. Identify the quadrants where tan(θ) is positive or negative.
4. Write the general solutions.

**Example 3:** Solve tan(θ) = 1.

1. Here, k = 1, which is valid.

2. The reference angle is θ_ref = 45° (since tan(45°) = 1).

3. Tan is positive in the first and third quadrants.

4. The general solutions are:

θ = 45° + 180°n and θ = 180° + 45° + 180°n.

Thus, θ = 45° + 180°n and θ = 225° + 180°n.

Multiple Angle Equations

Sometimes, you may encounter equations involving multiple angles, such as sin(2θ) = k or cos(3θ) = k. To solve these, you can use the following steps:

1. Use the appropriate trigonometric identity to rewrite the equation. For example, sin(2θ) = 2sin(θ)cos(θ).
2. Set the new equation equal to k.
3. Follow the same steps as above to find the solutions.

**Example 4:** Solve sin(2θ) = 1.

1. Rewrite using the identity: sin(2θ) = 2sin(θ)cos(θ).

2. Set 2sin(θ)cos(θ) = 1.

3. Rearranging gives sin(θ)cos(θ) = 1/2.

4. This equation can be solved by finding the angles where sin(θ) and cos(θ) yield this product.

5. The solutions can be calculated as:

θ = 15° + 180°n and θ = 75° + 180°n.

Using the Unit Circle

The unit circle is a valuable tool for solving trigonometric equations. The coordinates of points on the unit circle correspond to the values of sin(θ) and cos(θ). For example, the point (1, 0) corresponds to θ = 0°, and the point (0, 1) corresponds to θ = 90°.

When solving equations, you can find the angles by identifying the coordinates that give the desired sine or cosine value. This can help in visualising the solutions and understanding their periodic nature.

Check Your Solutions

After finding the general solutions, it is essential to check if they satisfy the original equation. Substitute the values back into the equation to ensure they hold true. This step is crucial, especially when dealing with multiple angles or transformations.

Summary

• Identify the type of trigonometric equation.
• Check the range of k for sine and cosine equations.
• Use reference angles and quadrants to find general solutions.
• Utilise identities for multiple angle equations.
• Check your solutions by substituting them back into the original equation.

Check Your Understanding

1. Solve sin(θ) = -0.5 for θ in the range [0°, 360°].
2. Find all solutions for cos(θ) = -1/2.
3. Determine the general solutions for tan(θ) = 0.
4. Solve sin(2θ) = 0 for θ in the range [0°, 360°].