# If #sectheta = a# and #sintheta < 0#, find the exact valaues of #sintheta#?

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I think the answer is like #-sqrt(a^2-1)#

Thanks in advance!!!

I think the answer is like

Thanks in advance!!!

See below

Squaring both sides

Then, transposing terms

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If sec(θ) = a and sin(θ) < 0, we can use the given information to find the exact value of sin(θ).

From the definition of secant and sine, we know that:

sec(θ) = 1/cos(θ) = a

From this, we can find the value of cos(θ):

cos(θ) = 1/sec(θ) = 1/a

Since sin(θ) < 0, θ must lie in the third or fourth quadrant, where sin is negative.

In the third or fourth quadrant, the cosine is negative. Since cos(θ) = 1/a is positive, θ must be in the fourth quadrant. Therefore, we can write:

cos(θ) = -√(1 - sin²(θ)) = 1/a

Solving for sin(θ):

-√(1 - sin²(θ)) = 1/a 1 - sin²(θ) = 1/a² sin²(θ) = 1 - 1/a² sin²(θ) = (a² - 1)/a² sin(θ) = ±√((a² - 1)/a²)

Since sin(θ) < 0, sin(θ) = -√((a² - 1)/a²).

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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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