Find the exact value of (sinθ) (cosθ) if cosθ > 0 and tanθ = −2/3
Draw a right triangle with legs 2 and 3, in QIV. Then
y = -2
x = -3
r = √13
sinθ = y/r
cosθ = x/r
I'm confused with this topic, can you please explain it some more
Really? Read about angles in the standard position.
tanθ < 0 in QII and QIV. You say cosθ is positive, so that means x is positive
Did you draw the triangle?
sinθ = y/r = -2/√13
cosθ = x/r = -3/√13
Note that sin^2θ + cos^2θ = 1, as required.
sin^2θ + cos^2θ = y^2/r^2 + x^2/r^2 = (x^2 + y^2)/r^2 = 1
OHH thank you so much, i understand it now
so then the exact value would just be 1 though?
no! The exact value of sinθ cosθ = (-2/√13)(-3/√13) = 6/13
To find the exact value of (sinθ) (cosθ) if cosθ > 0 and tanθ = −2/3, we can use the given information about cosθ and tanθ to determine the values of sinθ and cosθ separately.
First, we know that cosθ > 0, which means that the cosine function is positive. This tells us that θ lies in the first or fourth quadrant.
Second, we are given that tanθ = -2/3. Since tanθ is negative, we know that θ lies in the second or fourth quadrant.
To find the values of sinθ and cosθ, we can use the Pythagorean identity for tangent: tanθ = sinθ / cosθ. Substituting the given value of tanθ = -2/3, we have:
-2/3 = sinθ / cosθ
To solve this equation, we can cross multiply:
-2 * cosθ = 3 * sinθ
Next, we use the Pythagorean identity sin^2θ + cos^2θ = 1 to eliminate one variable. Since sin^2θ + cos^2θ = 1, we can rewrite sinθ in terms of cosθ:
sinθ = √(1 - cos^2θ)
Substituting this into the equation -2 * cosθ = 3 * sinθ, we get:
-2 * cosθ = 3 * √(1 - cos^2θ)
We can square both sides of the equation to eliminate the square root:
4 * cos^2θ = 9 * (1 - cos^2θ)
Simplifying, we have:
4 * cos^2θ = 9 - 9 * cos^2θ
Combining like terms:
13 * cos^2θ = 9
Dividing both sides by 13, we find:
cos^2θ = 9/13
Taking the square root of both sides, we have:
cosθ = ± √(9/13)
Since cosθ is positive (given in the problem statement), we take the positive square root:
cosθ = √(9/13)
Now that we have the value of cosθ, we can substitute it back into the equation -2 * cosθ = 3 * √(1 - cos^2θ) to find sinθ:
-2 * √(9/13) = 3 * √(1 - (√(9/13))^2)
Simplifying, we have:
-2 * √(9/13) = 3 * √(1 - 9/13)
-2 * √(9/13) = 3 * √(13/13 - 9/13)
-2 * √(9/13) = 3 * √(4/13)
-2 * √(9/13) = 3 * (2/√13)
-2 * √(9/13) = 6/√13
Thus, the exact value of (sinθ) (cosθ) is -6/√13.