Question

Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV

Accepted Answer

Identify the given information: $$\sin x = \frac{3}{5}$$ with $$x in $$quadrant I, and $$\cos y = \frac{24}{25}$$ with $$y in $$quadrant IV. Since $$x is in $$quadrant I, both $$\sin x$$ and $$\cos x$$ are positive. Since $$y is in $$quadrant IV, $$\cos y is $$positive and $$\sin y is $$negative. Find $$\cos x$$ using the Pythagorean identity $$\sin^2$ x + \cos^2$ x = 1. $$Substitute $$\sin x = \frac{3}{5} to $$get $$\cos x = \sqrt{1 - \left(\frac{3}{5}\$$right)^2$$}. $$Since $$x is in $$quadrant I, take the positive root. Find $$\sin y$$ using the Pythagorean identity $$\sin^2$ y + \cos^2$ y = 1. $$Substitute $$\cos y = \frac{24}{25} to $$get $$\sin y = -\sqrt{1 - \left(\frac{24}{25}\$$right)^2$$}. $$Since $$y is in $$quadrant IV, take the negative root. Use the angle sum and difference formulas to find the required values: 
- $$\sin(x + y) = \sin x \cos y + \cos x \sin y$$
- $$\cos(x - y) = \cos x \cos y + \sin x \sin y$$
- $$	an(x + y) = \frac{\sin(x + y)}{\cos(x + y)}$$, where $$\cos(x + y) = \cos x \cos y - \sin x \sin y. $$Determine the quadrant of $$x + y by $$analyzing the signs of $$\sin(x + y)$$ and $$\cos(x + y). $$Recall that:
- Quadrant I: $$\sin \u003e 0$$, $$\cos \u003e 0
- $$Quadrant II: $$\sin \u003e 0$$, $$\cos  0.$$

Use the given information to find sin(x + y), cos(x - y), tan(x + y), and the quadrant of x + y.
sin x = 3/5, cos y = 24/25, x in quadrant I, y in quadrant IV

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Key Concepts

Trigonometric Angle Sum and Difference Formulas

Determining Trigonometric Ratios from Quadrants

Finding Missing Trigonometric Values Using Pythagorean Identity