Final answer:
To find tan theta when cos theta = 3/5, we use the Pythagorean theorem to find sin theta, which is 4/5, then divide by cos theta to get tan theta = 4/3.
Step-by-step explanation:
If cos theta = 3/5, we need to find the value of tan theta. Given the identity tan theta = sin theta / cos theta, we must determine sin theta. By definition, for a right triangle, cos theta is the adjacent side over the hypotenuse. Hence, if cos theta = 3/5, the adjacent side (x) is 3 and the hypotenuse (h) is 5.
Using the Pythagorean theorem, which states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse (x² + y² = h²), we can solve for the opposite side (y). Since we know x and h, we find y = √(h² - x²) = √(5² - 3²) = √(25 - 9) = √16 = 4. So, the opposite side of our right triangle is 4.
Now, we know that sin theta is the opposite side over the hypotenuse (y/h), so sin theta = 4/5. Using the value of sin theta, we can find tan theta as tan theta = sin theta / cos theta = (4/5) / (3/5) = 4/3.
Therefore, tan theta = 4/3.