Angle θ lies in the second quadrant, and sin θ = 3/5.
Find cos θ and tan θ.
Draw the triangle.
sinθ = y/r
cosθ = x/r
tanθ = y/x
In QII,
y = 3
x = -4
r = 5
so,
cosθ = -4/5
tanθ = -3/4
To find the values of cos θ and tan θ, we can use the relationship between the trigonometric functions. We know that sin θ = opposite/hypotenuse, which means that the opposite side of the angle θ has a length of 3 and the hypotenuse has a length of 5.
Since angle θ lies in the second quadrant, the adjacent side will be negative, and we can use the Pythagorean theorem to find its length. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Let the adjacent side be represented by a. We have the following:
a^2 + 3^2 = 5^2
a^2 + 9 = 25
a^2 = 16
a = √16
a = -4 (since angle θ lies in the second quadrant)
Now, we can calculate cos θ and tan θ using the values we found:
cos θ = adjacent/hypotenuse = -4/5
tan θ = opposite/adjacent = 3/-4
Therefore, cos θ = -4/5 and tan θ = -3/4.
To find cos θ and tan θ, we'll use the Pythagorean Identity and the definition of cosine and tangent.
Given that sin θ = 3/5, we can determine the value of cos θ using the Pythagorean Identity:
sin^2 θ + cos^2 θ = 1
(3/5)^2 + cos^2 θ = 1
9/25 + cos^2 θ = 1
cos^2 θ = 1 - 9/25
cos^2 θ = 25/25 - 9/25
cos^2 θ = 16/25
Taking the square root of both sides:
cos θ = ± √(16/25)
Since θ lies in the second quadrant, the cosine value will be negative:
cos θ = - √(16/25)
cos θ = - 4/5
Next, we can determine the value of tan θ by using the definition of tangent:
tan θ = sin θ / cos θ
tan θ = (3/5) / (-4/5)
tan θ = 3/5 * (-5/4)
tan θ = -3/4
Therefore, cos θ = -4/5 and tan θ = -3/4.