Write the first trigonometric function in terms of the second for θ in the given quadrant.
tan(θ), cos(θ); θ in Quadrant IV
I believe to start this is 1/cos^2*theta. Am I write? If not, can you please help me?!
sin theta = + OR - sqroot ( 1 - cos ^ 2 theta )
tan theta = sin theta / cos theta
tan theta = + OR - sqroot ( 1 - cos ^ 2 theta ) / cos theta
In Quadrant IV tangent are negative so :
tan theta = - sqroot ( 1 - cos ^ 2 theata ) / cos theta
Well, it seems like you have a bit of a mix-up there! Let me help you out with the correct trigonometric function for θ in Quadrant IV.
In this case, since θ is in Quadrant IV, we know that both x and y values are negative.
The cosine function, cos(θ), is positive in Quadrant IV since x is negative. However, the tangent function, tan(θ), is negative in Quadrant IV since both x and y are negative.
So, the correct trigonometric function for θ in Quadrant IV is:
tan(θ) = -sin(θ)/cos(θ).
Remember, it's always good to double-check the signs when working with trigonometric functions in different quadrants. Keep up the good work!
In Quadrant IV, the value of cosine (cos) is positive, while the value of tangent (tan) is negative. Therefore, we need to determine the relationship between the two functions in this quadrant.
The tangent function is defined as the ratio of the sine function to the cosine function:
tan(θ) = sin(θ) / cos(θ)
To express tan(θ) in terms of cos(θ) in Quadrant IV, we can use the reciprocal identity for cosine:
cos(θ) = 1 / sec(θ)
Substituting this into the expression for tangent gives:
tan(θ) = sin(θ) / (1 / sec(θ))
Recall that the secant function is the reciprocal of cosine:
sec(θ) = 1 / cos(θ)
Replacing sec(θ) in the expression, we have:
tan(θ) = sin(θ) / (1 / (1 / cos(θ)))
Simplifying further, we get:
tan(θ) = sin(θ) / (1 / (1 / cos(θ)))
= sin(θ) / (1 / cos(θ))
= sin(θ) * cos(θ)
Therefore, the first trigonometric function (tan(θ)) in terms of the second (cos(θ)) in Quadrant IV is sin(θ) * cos(θ).
So, the correct expression for tan(θ) in terms of cos(θ) in Quadrant IV is tan(θ) = sin(θ) * cos(θ).
To find the first trigonometric function in terms of the second for θ in Quadrant IV (where cos(θ) is positive and tan(θ) is negative), we need to use the trigonometric identity relating tangent and cosine.
In Quadrant IV, the x-coordinate is positive and the y-coordinate is negative. Since tangent is defined as the ratio of sine to cosine, we can express tan(θ) as sin(θ) / cos(θ).
Now, we can use the Pythagorean identity to find sin(θ):
sin²(θ) + cos²(θ) = 1
Rearranging the equation, we get:
sin²(θ) = 1 - cos²(θ)
Taking the square root of both sides:
sin(θ) = √(1 - cos²(θ))
Now, we can substitute this expression for sin(θ) into our expression for tan(θ):
tan(θ) = sin(θ) / cos(θ)
tan(θ) = √(1 - cos²(θ)) / cos(θ)
Therefore, the first trigonometric function, tan(θ), in terms of the second, cos(θ), for θ in Quadrant IV, is √(1 - cos²(θ)) / cos(θ).