Question

If sin x sec x = -1 and x lies in the second quadrant, find sin x and sec x.

Answer 1

the angle is 135 degrees & sin135=0.0884
sec 135=-1.0884

Answer 2

Answer:

$sin x=\frac{1}{\sqrt{2} } , sec x=-\sqrt{2}$

Step-by-step explanation:

Given:

sin x sec x = -1

To find:

sin x and sec x

Solution:

Trigonometric functions can be divided into six main categories: sine (sin), cosine (cos), secant (sec), cosecant (cosec), tangent (tan), and cotangent (cot). The relationship between the ratios of the various sides of a right-angle triangle is represented by the six different types of trigonometric functions. As the ratios of the x and y coordinates of the circle of Radius 1 maintain contact with the angles in standard locations, these trigonometric functions can also be referred to as circular functions.

sin x sec x = -1

$sinx(\frac{1}{cosx} )=-1\\sin^{2}x=cos^{2}x \\sin^{2}x=1-sin^{2}x\\sin^{2}x=\frac{1}{2} \\\\sinx=+\frac{1}{\sqrt{2}} or -\frac{1}{\sqrt2} } \\sin x=\frac{1}{\sqrt{2} } , sec x=-\sqrt{2}$

We know that x lies in second quadrant ( given) so,

$x=\frac{3\pi }{4} \\sinx=sin\frac{3\pi }{4}=\frac{1}{\sqrt{2} } \\secx=sec\frac{3\pi }{4}=-\sqrt{2}$

Hence the correct answer is $sin x=\frac{1}{\sqrt{2} } , sec x=-\sqrt{2}$

sin theta( 1+tan theta) + cos theta(1+ cot theta) = sec theta + cosec theta

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