If tan(theta)=7/13 and cos(theta)<0, what is the exact value of sin(theta)?
Tan is positive in the first quadrant, and the third quadrant.
cos is positive in first and fourth quadrant.
Which makes Theta in third quadrant (tan positive, cos negative). Sine therefore must also be negative.
Draw the triangle, legs 7 and 13, determine the hypotenuse, then sine.
To find the exact value of sin(theta), we need to use the given information about theta.
From the equation tan(theta) = 7/13, we can conclude that the tangent of theta is positive since 7/13 is positive. Tangent is positive in the first and third quadrants.
Next, we are given that cos(theta) < 0. Since cosine is negative, we know that theta is in the second or third quadrant.
Combining these two pieces of information, we can determine that theta must be in the third quadrant, where tangent is positive and cosine is negative.
Since we are dealing with a right triangle in the third quadrant, we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
We already know that cos(theta) < 0, so we can express it as cos(theta) = -sqrt(1 - sin^2(theta)).
Substituting this into the equation, we get:
sin^2(theta) + (-sqrt(1 - sin^2(theta)))^2 = 1
Simplifying further:
sin^2(theta) + (1 - sin^2(theta)) = 1
Expanding and simplifying again:
sin^2(theta) + 1 - sin^2(theta) = 1
1 = 1
This equation is true for all values of sin(theta). Therefore, there is no restriction on the exact value of sin(theta) given the conditions provided.
In conclusion, the exact value of sin(theta) could be any real number.