Use trigonometric identities to write sin X in terms of cosx Choose the correct expression equal to sin x sin x = 1- cos X sin x= +v1- cosx sin x = (1 – cosx)(1 cos X) sin x= cos
The Correct Answer and Explanation is:
To express sin(x) in terms of cos(x) using trigonometric identities, we start from the Pythagorean identity:sin2(x)+cos2(x)=1\sin^2(x) + \cos^2(x) = 1sin2(x)+cos2(x)=1
This can be rearranged to solve for sin²(x):sin2(x)=1−cos2(x)\sin^2(x) = 1 – \cos^2(x)sin2(x)=1−cos2(x)
Now, take the square root of both sides:sin(x)=1−cos2(x)\sin(x) = \sqrt{1 – \cos^2(x)}sin(x)=1−cos2(x)
However, because square roots can be positive or negative, the sign depends on the quadrant in which angle x lies. If x is in the first or second quadrant, then sin(x) is positive. If x is in the third or fourth quadrant, then sin(x) is negative.
So the correct identity is:sin(x)=±1−cos2(x)\sin(x) = \pm \sqrt{1 – \cos^2(x)}sin(x)=±1−cos2(x)
Now look at your answer choices:
- sin x = 1 – cos x → Incorrect. This is not a valid trigonometric identity.
- sin x = +√(1 – cos x) → Incorrect. It should be cos²(x) inside the square root, not cos(x).
- sin x = (1 – cos x)(1 + cos x) → This equals 1 – cos²(x), which is sin²(x), not sin(x).
- sin x = cos → Incorrect. Sin(x) is not equal to cos(x) in general.
So the correct answer should be:sin(x)=±1−cos2(x)\sin(x) = \pm \sqrt{1 – \cos^2(x)}sin(x)=±1−cos2(x)
Even though it is not listed exactly among the choices, it is important to recognize this as the correct expression derived from identities. It helps solve many problems in trigonometry, especially in calculus or geometry, where expressing one trig function in terms of another is often necessary.
In summary, we used the Pythagorean identity to isolate sin²(x) and then took the square root to solve for sin(x). The sign depends on the quadrant of angle x. This identity is fundamental and commonly used in trigonometric simplification and solving equations.
