Question

Find tan theta given that sin theta = -4/5 and cos theta > 0

Answer 1

To find tan theta with sin theta = -4/5 and cos theta > 0, we determine that theta is in the fourth quadrant and use the Pythagorean identity. Computing cos theta and dividing sin theta by cos theta, we find that tan theta equals -4/3.

Step-by-step explanation:

To find the value of tan theta when given that sin theta = -4/5 and cos theta > 0, we must first understand which quadrant theta lies in. Since the sine value is negative and the cosine value is positive, theta must be in the fourth quadrant. In this quadrant, tangent values are negative.

Next, we apply the Pythagorean identity which states that sin2(theta) + cos2(theta) = 1.

Using sin theta = -4/5, we can find cos theta:

cos theta = sqrt(1 - sin2(theta)) = sqrt(1 - (-4/5)2) = sqrt(1 - 16/25) = sqrt(9/25) = 3/5.

Note that we take the positive value for cosine since cos theta > 0.

Finally, we can calculate tan theta as tan theta = sin theta / cos theta = (-4/5) / (3/5) = -4/3.

Answer 2

Given:
sin theta = -4/5
cos theta >0
Since sin is negative and cos is positive, theta lies in the 4th quadrant. All trigonometric ratios, except cos and sec are negative in the fourth quadrant.
Therefore, tan is also negative.
sin theta = Perpendicular/Hypotenuse
Therefore, perpendicular = 4 units and hypotenuse =5 units.
According to Pythagorean theorem, in a right triangle, the sum of squares of the legs of a right triangle is equal to the square of the hypotenuse.
So, in a right triangle ABC, if AB is the perpendicular, BC is the adjacent side and AC is the hypotenuse, then
AC^2 = AB^2 + BC^2
BC^2=AC^2-AB^2
BC^2= 5^2 - 4^2
=25 - 16
BC ^2 = 9
BC = 3 units
Adjacent side = 3 units
We know that tan theta = perpendicular/Adjacent
=4/3
since tan is negative in the 4th quadrant,
tan theta =-4/3