Final answer:
To find all the other trigonometric values when sin(theta) = 1/3, we calculate the adjacent side using the Pythagorean theorem and then find cosine, tangent, secant, cosecant, and cotangent based on the ratios of a right triangle's sides.
Step-by-step explanation:
Given that sin(theta) = 1/3, we can use trigonometric identities and the Pythagorean theorem to find the other trigonometric values of theta. In a right triangle with angle theta, the sine is defined as the ratio of the opposite side (y) to the hypotenuse (h). Therefore, if sin(theta) = 1/3, we can consider a right triangle where the opposite side y is 1 and the hypotenuse h is 3.
To find the cosine of theta, we need to determine the length of the adjacent side x. According to the Pythagorean theorem, we have:
- x² + y² = h²
- x² + 1² = 3²
- x² = 9 - 1
- x² = 8
- x = √8
Since x = √8, the cosine of theta is cos(theta) = adjacent/hypotenuse = √8 / 3.
Next, we find the tangent (tan(theta)), which is the ratio of the opposite to the adjacent side, hence tan(theta) = opposite/adjacent = 1 / √8, which can be rationalized to √8 / 8.
For secant (sec(theta)), which is the reciprocal of cosine, we get sec(theta) = hypotenuse/adjacent = 3 / √8, which can be rationalized to 3√8 / 8.
The cosecant (csc(theta)), the reciprocal of sine, is csc(theta) = hypotenuse/opposite = 3 / 1 = 3.
Finally, the cotangent (cot(theta)), the reciprocal of tangent, is cot(theta) = adjacent/opposite = √8 / 1 = √8.