How do you find the five remaining trigonometric function satisfying #sintheta=3/8#, #costheta<0#?
Now that you have all side lengths, simply plug them in to the remaining trigonometric functions:
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Given ( \sin(\theta) = \frac{3}{8} ) and ( \cos(\theta) < 0 ), you can use the Pythagorean identity ( \sin^2(\theta) + \cos^2(\theta) = 1 ) to find ( \cos(\theta) ).
[ \sin^2(\theta) + \cos^2(\theta) = 1 ] [ \left(\frac{3}{8}\right)^2 + \cos^2(\theta) = 1 ] [ \frac{9}{64} + \cos^2(\theta) = 1 ] [ \cos^2(\theta) = 1  \frac{9}{64} ] [ \cos^2(\theta) = \frac{64}{64}  \frac{9}{64} ] [ \cos^2(\theta) = \frac{55}{64} ]
Since ( \cos(\theta) < 0 ), ( \cos(\theta) ) must be negative. Therefore, ( \cos(\theta) = \sqrt{\frac{55}{64}} = \frac{\sqrt{55}}{8} ).
Now, you can find the remaining trigonometric functions using the definitions:

( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} ) [ \tan(\theta) = \frac{\frac{3}{8}}{\frac{\sqrt{55}}{8}} = \frac{3}{\sqrt{55}} ]

( \cot(\theta) = \frac{1}{\tan(\theta)} ) [ \cot(\theta) = \frac{1}{\frac{3}{\sqrt{55}}} = \frac{\sqrt{55}}{3} ]

( \sec(\theta) = \frac{1}{\cos(\theta)} ) [ \sec(\theta) = \frac{1}{\frac{\sqrt{55}}{8}} = \frac{8}{\sqrt{55}} ]

( \csc(\theta) = \frac{1}{\sin(\theta)} ) [ \csc(\theta) = \frac{1}{\frac{3}{8}} = \frac{8}{3} ]
Therefore, the five remaining trigonometric functions satisfying ( \sin(\theta) = \frac{3}{8} ) and ( \cos(\theta) < 0 ) are:
[ \tan(\theta) = \frac{3}{\sqrt{55}} ] [ \cot(\theta) = \frac{\sqrt{55}}{3} ] [ \sec(\theta) = \frac{8}{\sqrt{55}} ] [ \csc(\theta) = \frac{8}{3} ]
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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