Hi there! Can anyone help solve this? :) Given tan theta = p and that theta is an acute angle, find sec 2 theta and cot (90-2theta)

Answer 1

#sec theta = 1/cos theta = \sqrt{p^2 + 1 }#

# cot (90^circ - 2 theta) = {2p}/{1-p^2}#

#tan theta = p = p/1 #
That's a right triangle with opposite #p# and adjacent #1#, so hypotenuse #\sqrt{p^2+1}.#
So #cos theta = 1/{pm \sqrt{p^2+1} }#
We choose the positive square root because we're told #theta# is acute.
#sec theta = 1/cos theta = \sqrt{p^2 + 1 }#
# cot (90^circ - 2 theta) #
# = tan(2 theta ) #
# = {2 tan theta}/{1 - tan^2 theta } #
# = {2p}/{1-p^2}#
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Answer 2

Sure! Let's solve the given expressions step by step:

  1. Find sec(2θ): Given that ( \tan(\theta) = p ), we know that ( \sec^2(\theta) = 1 + \tan^2(\theta) ). Now, we need to find ( \sec(2\theta) ). Using the double angle identity for secant: ( \sec(2\theta) = \frac{1}{\cos(2\theta)} ). Using the double angle identity for cosine: ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) ). Given that ( \tan(\theta) = p ), we can find ( \sin(\theta) ) and ( \cos(\theta) ) using the relation ( \tan^2(\theta) + 1 = \sec^2(\theta) ). Then, ( \sin(\theta) = \frac{p}{\sqrt{p^2 + 1}} ) and ( \cos(\theta) = \frac{1}{\sqrt{p^2 + 1}} ).

Now, substitute ( \sin(\theta) ) and ( \cos(\theta) ) into the expression for ( \cos(2\theta) ), and then find ( \sec(2\theta) ) using ( \sec(2\theta) = \frac{1}{\cos(2\theta)} ).

  1. Find ( \cot(90 - 2\theta) ): Using the complementary angle identity for cotangent: ( \cot(90 - 2\theta) = \tan(2\theta) ). Using the double angle identity for tangent: ( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} ). Given that ( \tan(\theta) = p ), we can find ( \tan^2(\theta) ).

Now, substitute ( \tan(\theta) = p ) into the expression for ( \tan^2(\theta) ), then find ( \tan(2\theta) ) using the relation ( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} ).

These steps will help you find the values of ( \sec(2\theta) ) and ( \cot(90 - 2\theta) ).

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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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