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Given cos A = 3/ sqrt 10
and cot A = -3, determine the value of sin A in radical form.

User Joss
by
7.5k points

2 Answers

1 vote

Final Answer:

The value of sin A in radical form is:


\[ \sin A = -(√(10))/(10) \]

Step-by-step explanation:

The given information includes
\( \cos A = (3)/(√(10)) \) and \( \cot A = -3 \). To determine \( \sin A \), we can use the identity
\( \cot A = (\cos A)/(\sin A) \). Substituting the given values, we get
\( -3 = ((3)/(√(10)))/(\sin A) \).

To isolate
\( \sin A \), we multiply both sides by \( \sin A \)and simplify, resulting in
\( \sin A = -(√(10))/(10) \). The negative sign indicates that
\( \sin A \)is negative, placing the angle in the third quadrant where both sine and cosine are negative.

In the third quadrant, the cosine is negative and the sine is also negative. Therefore,
\( \sin A = -(√(10))/(10) \). This means that the ratio of the opposite side to the hypotenuse in the right triangle formed by angle
\( A \) is \(-(√(10))/(10)\).

User Carlesba
by
7.7k points
4 votes
We can use the definitions of cosine and cotangent to find the value of sin A.

cos A = adjacent/hypotenuse = 3/√10

cot A = adjacent/opposite = -3

We can use the Pythagorean theorem to find the value of the opposite side of the triangle:

a^2 + b^2 = c^2

3^2 + b^2 = 10

b^2 = 1

b = 1

So the opposite side is 1.

Now we can use the definition of cotangent to find the adjacent side:

cot A = adjacent/opposite

-3 = adjacent/1

adjacent = -3

Now we can use the Pythagorean theorem to find the value of the hypotenuse:

a^2 + b^2 = c^2

(-3)^2 + 1^2 = c^2

9 + 1 = c^2

c^2 = 10

c = √10

So the hypotenuse is √10.

Finally, we can use the definition of sine to find the value of sin A:

sin A = opposite/hypotenuse = 1/√10

Therefore, the value of sin A in radical form is 1/√10.
User Orlando Herrera
by
8.0k points

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