if 3sin theta =2 and cos greater than 0,with the aid of a diagram determine the value of:
a) tan theta
b) sin theta/cos theta
c) sin^2 theta + cos^2 theta
you have
sinθ = 2/3
if sin and cos are both positive, then θ is in QI.
So, now draw the triangle.
y = 2
r = 3, so
x = √5
Now just use those values and the fact that
sinθ = y/r
cosθ = x/r
tanθ = y/x
to answer the questions.
The last one should be easy, since it is always the same value.
To solve this equation, let's start by drawing a right triangle with an angle θ, where the opposite side is labeled '3' and the hypotenuse is labeled '2'. Since cos is greater than 0, we know that the adjacent side is positive.
Using the Pythagorean theorem, we can find the length of the adjacent side as follows:
a^2 + 3^2 = 2^2
a^2 + 9 = 4
a^2 = -5 (This is not possible because it would involve taking the square root of a negative number. So, there is no triangle that fits the given conditions.)
Therefore, we cannot determine the values of a) tan(theta), b) sin(theta)/cos(theta), and c) sin^2(theta) + cos^2(theta) since there is no valid right triangle based on the given conditions.
To determine the value of various trigonometric functions (such as tan theta, sin theta/cos theta, and sin^2 theta + cos^2 theta) when given an equation involving sin theta and cos theta, we first need to find the value of theta.
Given the equation 3sin theta = 2 and the condition cos theta > 0, we can use the equation 3sin theta = 2 to solve for sin theta.
To do this, divide both sides of the equation by 3:
sin theta = 2/3
Now we know the value of sin theta, but we need to find the value of theta itself. To obtain the value of theta, we can use the inverse sine function (sin^-1) or the arcsin function.
Using a calculator or a table of trigonometric values, find the value of theta such that sin theta = 2/3 and cos theta > 0. Let's assume that theta is in the first quadrant, where both sin and cos values are positive.
a) To determine the value of tan theta:
Using the calculated value of theta, use the tangent function (tan) to find tan theta:
tan theta = sin theta / cos theta
b) To determine the value of sin theta / cos theta:
Divide the value of sin theta by the value of cos theta.
c) To determine the value of sin^2 theta + cos^2 theta:
Use the Pythagorean identity for sine and cosine:
sin^2 theta + cos^2 theta = 1
By substituting the values of sin theta and cos theta obtained earlier into the above equation, you can find the value of sin^2 theta + cos^2 theta.
Remember to use a calculator or a table of trigonometric values to find the specific values of theta and the trigonometric functions. A diagram can help visualize the situation, but it may not be sufficient to find the actual values.