Question

Evaluate each expression without using a calculator.sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

Accepted Answer

Recognize that the expression is of the form $$\sin(\sin$$^{-1}$$ x + 	an$$^{-1}$$ y)$$, where $$x = \frac{1}{2}$$ and $$y = -3. $$Our goal is to simplify this without a calculator. Use the sine addition formula: $$\sin(A + B) = \sin A \cos B + \cos A \sin B. $$Here, let $$A = \sin$$^{-1}$$ \frac{1}{2}$$ and $$B = 	an$$^{-1}$$ (-3)$$, so rewrite the expression as $$\sin A \cos B + \cos A \sin B. $$Find $$\sin A$$ and $$\cos A$$: Since $$A = \sin$$^{-1}$$ \frac{1}{2}$$, we know $$\sin A = \frac{1}{2}. To $$find $$\cos A$$, use the Pythagorean identity $$\cos A = \sqrt{1 - \sin^2$ A} = \sqrt{1 - \left(\frac{1}{2}\$$right)^2$$}. $$Find $$\sin B$$ and $$\cos B$$: Since $$B = 	an$$^{-1}$$ (-3)$$, let $$	an B = -3 = \frac{\sin B}{\cos B}. $$Represent this as a right triangle with opposite side $$-3$$ and adjacent side $$1$$, then find the hypotenuse $$\sqrt{1^2$ + (-3)^2$} = \sqrt{10}. $$Thus, $$\sin B = \frac{-3}{\sqrt{10}}$$ and $$\cos B = \frac{1}{\sqrt{10}}. $$Substitute all values back into the sine addition formula: $$\sin A \cos B + \cos A \sin B = \left(\frac{1}{2}ight) \left(\frac{1}{\sqrt{10}}ight) + \left(\sqrt{1 - \left(\frac{1}{2}\$$right)^2$$}ight) \left(\frac{-3}{\sqrt{10}}ight). $$This expression can now be simplified step-by-step.

Evaluate each expression without using a calculator.
sin (sin⁻¹ 1/2 + tan⁻¹ (-3))

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Concetti chiave

Inverse Trigonometric Functions

Trigonometric Addition Formulas

Right Triangle Interpretation of Inverse Tangent