If sin(θ)=8/10, 0≤θ≤π/2, then
cos(theta)=
tan(theta)=
Sec(theta)=
Please explain how to do this
Well, if sin(θ) = 8/10, then we can figure out the value of cos(θ) using the Pythagorean identity. Since sin(θ) represents the ratio of the length of the side opposite θ to the length of the hypotenuse, and we're given that sin(θ) = 8/10, we can let the opposite side be 8 and the hypotenuse be 10.
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can solve for the adjacent side, which we'll call "a" (corresponding to cos(θ)).
So, a^2 + 8^2 = 10^2
a^2 + 64 = 100
a^2 = 100 - 64
a^2 = 36
a = √36
a = 6
Therefore, cos(θ) = 6/10, which simplifies to 3/5.
Now, let's find tan(θ). We know that tan(θ) = sin(θ)/cos(θ). So, tan(θ) = (8/10)/(3/5). Simplifying, we get tan(θ) = 8/6, which simplifies further to 4/3.
Finally, we can determine the value of sec(θ), which is the reciprocal of the cosine. So, sec(θ) = 1/cos(θ). Plugging in the value of cos(θ) = 3/5, we find that sec(θ) = 1/(3/5), which simplifies to 5/3.
Hope that clears things up! Remember, math is just like clowning around, but with numbers instead of balloons.
To find the values of cosine (cos), tangent (tan), and secant (sec) of theta (θ) given that sin(θ) is equal to 8/10 with a domain of 0 ≤ θ ≤ π/2, you can use the Pythagorean identity and the definitions of these trigonometric functions.
Step 1:
We know that sin(θ) = opposite/hypotenuse. Since sin(θ) = 8/10, we can set the opposite side of the right triangle formed by θ to be 8 and the hypotenuse to be 10.
Step 2:
To find the adjacent side, we'll use the Pythagorean theorem: a^2 + b^2 = c^2, where a and b are the lengths of the two sides of a right triangle and c is the length of the hypotenuse.
In this case, we have a = adjacent side, b = opposite side, and c = hypotenuse. We substitute the known values: a^2 + 8^2 = 10^2.
Simplifying the equation gives us a^2 + 64 = 100.
Subtracting 64 from both sides, we have a^2 = 36.
Taking the square root of both sides, we get a = 6.
Step 3:
With the lengths of the two sides of the right triangle, we can find the values of cosine (cos), tangent (tan), and secant (sec) using the definitions:
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent
sec(θ) = 1/cos(θ)
Plugging the values we found into these formulas:
cos(θ) = 6/10 = 3/5
tan(θ) = 8/6 = 4/3
sec(θ) = 1/cos(θ) = 1/(3/5) = 5/3
So, cos(θ) = 3/5, tan(θ) = 4/3, and sec(θ) = 5/3 for 0 ≤ θ ≤ π/2 given that sin(θ) = 8/10.
To find the values of cos(theta), tan(theta), and sec(theta) given sin(theta) = 8/10 and 0≤theta≤π/2, you can use the following trigonometric identities:
1. cos(theta) = √(1 - sin^2(theta))
2. tan(theta) = sin(theta)/cos(theta)
3. sec(theta) = 1/cos(theta)
First, let's find the value of cos(theta):
Using the given value of sin(theta) = 8/10, we can find cos(theta) using the Pythagorean Identity sin^2(theta) + cos^2(theta) = 1.
Substituting sin(theta) = 8/10, we get:
(8/10)^2 + cos^2(theta) = 1
64/100 + cos^2(theta) = 100/100
cos^2(theta) = 36/100
cos(theta) = √(36/100)
cos(theta) = 6/10
cos(theta) = 3/5
Therefore, cos(theta) = 3/5.
Next, let's find the value of tan(theta):
Using the given value of sin(theta) = 8/10 and cos(theta) = 3/5, we can find tan(theta) using the identity tan(theta) = sin(theta)/cos(theta).
Substituting sin(theta) = 8/10 and cos(theta) = 3/5, we get:
tan(theta) = (8/10)/(3/5)
tan(theta) = (8/10) * (5/3)
tan(theta) = (8*5)/(10*3)
tan(theta) = 40/30
tan(theta) = 4/3
Therefore, tan(theta) = 4/3.
Lastly, let's find the value of sec(theta):
Using the previously found value of cos(theta) = 3/5, we can find sec(theta) using the identity sec(theta) = 1/cos(theta).
Substituting cos(theta) = 3/5, we get:
sec(theta) = 1/(3/5)
sec(theta) = 1 * (5/3)
sec(theta) = 5/3
Therefore, sec(theta) = 5/3.
Recall that
sinθ = y/r
cosθ = x/r
tanθ = y/x
you have a triangle in QI where
y=8
x=6
r=10