Please help me.Iam stuck. Find all the possible values of theta between 0 degree and 360 degree for the following quadratic equation. State the quadrants in which each angle lie. 8 tan^2 theta + 2 tan theta=15
To find the possible values of theta for the quadratic equation 8tan^2(theta) + 2tan(theta) = 15, we can start by rearranging the equation in a standard quadratic form: 8tan^2(theta) + 2tan(theta) - 15 = 0.
Let's solve this quadratic equation step by step:
Step 1: Factor the equation, if possible.
Since factoring a quadratic equation involving trigonometric functions can be challenging, we will use the quadratic formula instead.
Step 2: Apply the quadratic formula.
The quadratic formula states that for any equation of the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a).
Applying this formula to our equation, where a = 8, b = 2, and c = -15, we can calculate the solutions for tan(theta):
tan(theta) = [-(2) ± √((2)^2 - 4(8)(-15))] / (2(8))
= [-2 ± √(4 + 480)] / 16
= [-2 ± √484] / 16
= (-2 ± 22) / 16.
This yields two possible solutions:
Case 1: tan(theta) = (20 / 16) = 5/4
Case 2: tan(theta) = (-24 / 16) = -3/2.
Step 3: Find the corresponding angles.
To find the values of theta corresponding to the tan(theta) solutions, we need to use an inverse trigonometric function, in this case, the arctan or inverse tangent function (tan^(-1)).
Case 1: For tan(theta) = 5/4, we find the corresponding angle theta by taking the inverse tangent (arctan) of 5/4:
theta = arctan(5/4).
Using a calculator or trigonometric tables, we find that the principal value of arctan(5/4) is approximately 51.34 degrees. However, the question specifies that we need the values of theta between 0 and 360 degrees.
To find the other values of theta in this range, we need to add or subtract multiples of 180 degrees, as tan(theta) values repeat every 180 degrees due to the periodic nature of the tangent function.
Therefore, the values of theta for tan(theta) = 5/4 are:
theta = 51.34 degrees + k * 180 degrees, where k is an integer.
Case 2: For tan(theta) = -3/2, we similarly take the inverse tangent (arctan) of -3/2:
theta = arctan(-3/2).
Using a calculator or trigonometric tables, we find that the principal value of arctan(-3/2) is approximately -56.31 degrees.
To find the other values of theta in the range of 0 to 360 degrees, we again add or subtract multiples of 180 degrees:
theta = -56.31 degrees + k * 180 degrees, where k is an integer.
Step 4: Identify the quadrants in which each angle lies.
To determine the quadrants in which each angle lies, we need to consider the signs of the tangent function in each quadrant:
- In the first quadrant (0° to 90°), the tangent function is positive.
- In the second quadrant (90° to 180°), the tangent function is negative.
- In the third quadrant (180° to 270°), the tangent function is positive.
- In the fourth quadrant (270° to 360°), the tangent function is negative.
Using this information, we can determine the quadrants in which each angle lies:
For Case 1 (tan(theta) = 5/4), the angle theta = 51.34 degrees lies in the first quadrant (0° to 90°).
For Case 2 (tan(theta) = -3/2), the angle theta = -56.31 degrees lies in the third quadrant (180° to 270°).
Therefore, the possible values of theta between 0 and 360 degrees for the equation 8tan^2(theta) + 2tan(theta) = 15 are:
- First quadrant: theta = 51.34 degrees + k * 180 degrees, where k is an integer.
- Third quadrant: theta = -56.31 degrees + k * 180 degrees, where k is an integer.