find two values of theta, zero degrees is than or equal to theta which is less than 360 degrees, that satisfy the given trigonometric equation. tan theta equals radical 3.
30 60 90 triangle is 1, sqrt 3 , 2
cos 60 = 1/2
sin 60 = sqrt 3 /2
tan 60 degrees = sqrt 3 / 1
so 60 degrees
tan is also + in quadrant 3
so
60 + 180 = 240 degrees
To find the values of theta that satisfy the trigonometric equation tan(theta) = √3, we can use the inverse tangent function or arctan on both sides of the equation. The inverse tangent function will help us find the angle whose tangent is equal to √3.
arctan(tan(theta)) = arctan(√3)
This simplifies to:
theta = arctan(√3)
We can use a calculator or a trigonometric table to find the principal value of arctan(√3). The principal value of arctan(√3) is 60 degrees since it is the angle within the range of -90 to 90 degrees whose tangent is √3.
So, one possible value of theta is 60 degrees.
To find another value of theta, we can use the property of the tangent function that it has a periodicity of 180 degrees. This means we can add or subtract any multiple of 180 degrees to the principal value to get additional solutions.
Next, we can add 180 degrees to the principal value:
theta = 60 degrees + 180 degrees = 240 degrees
So, another possible value of theta is 240 degrees.
Therefore, the two values of theta that satisfy the given trigonometric equation tan(theta) = √3 are 60 degrees and 240 degrees.