CrossFit415 said:Cos theta = -12/13 . And (-) since it's in quadrant IV.
To solve for θ, we can use the inverse trigonometric functions, specifically arcsine and arccosine. First, we can rewrite the given equations as sin θ = -5/13 and cos θ = -(√194)/13. Then, we can take the inverse sine of both sides of the first equation to get θ = arcsin(-5/13). Similarly, we can take the inverse cosine of both sides of the second equation to get θ = arccos(-(√194)/13). This will give us two possible values for θ.
Since the sine and cosine functions have a period of 2π, there are multiple values of θ that satisfy the given equations. To find all possible solutions, we need to consider the unit circle and use reference angles to determine the other values of θ that satisfy the equations.
The reference angle is the acute angle between the terminal side of θ and the x-axis. To determine the reference angle, we can use the given values of sine and cosine to find the corresponding coordinates on the unit circle. Then, we can use the Pythagorean theorem to find the length of the hypotenuse and the reference angle can be found using trigonometric ratios.
Yes, most scientific calculators have built-in functions for inverse trigonometric functions, which can be used to solve for θ. However, it is important to make sure that the calculator is in the correct mode (degrees or radians) and to check for multiple solutions.
To check your solutions, you can substitute the values of θ into the original equations and see if they satisfy the equations. You can also use a graphing calculator to graph the given equations and see if the coordinates of the intersection points match with your solutions for θ.