How do you find the five remaining trigonometric function satisfying #sintheta=3/8#, #costheta<0#?

Answer 1

#cosΘ=(-√55)/8#
#tanΘ=(-3√55)/55#
#cscΘ=8/3#
#secΘ=(-8√55)/55#
#cotΘ=(-√55)/3#

Since #sin# is positive and #cos# is less than 0, it means that this triangle is in quadrant 2.
Use Pythagorean's theorem (#a^2+b^2=c^2#) and plug in your given values: #a=3# and #c=8# and solve.
#3^2+b^2=8^2# → #9+b^2=64# → #b^2=55# → #b=-√55# (it is negative because it is in quadrant 2 and the #cos# is the x value of the triangle)

Now that you have all side lengths, simply plug them in to the remaining trigonometric functions:

#cosΘ=(-√55)/8# #tanΘ=3/(-√55)=(-3√55)/55# #cscΘ=8/3# #secΘ=8/(-√55)=(-8√55)/55# #cotΘ=(-√55)/3#
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Answer 2

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:

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

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

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

  4. ( \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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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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