Find theta if sin(theta+36°)=cos theta where theta +36 is an acute angle
To find the value of theta, we can start by using the identity that relates sine and cosine:
sin^2(theta) + cos^2(theta) = 1
Since sin(theta+36°) = cos(theta), we can square both sides of this equation:
(sin(theta+36°))^2 = (cos(theta))^2
Expanding the left side of the equation using the trigonometric identity for sine:
(sin(theta)cos(36°) + cos(theta)sin(36°))^2 = (cos(theta))^2
Simplifying further by using the trigonometric identities for cosine and sine of complementary angles:
[(sin(theta)cos(36°) + cos(theta)sin(36°))^2] + (sin(theta))^2 = (cos(theta))^2
Expanding and simplifying the left side of the equation:
[sin^2(theta)cos^2(36°) + 2sin(theta)cos(theta)sin(36°)cos(36°) + cos^2(theta)sin^2(36°)] + (sin(theta))^2 = (cos(theta))^2
Using the trigonometric identities for sine and cosine squared:
[sin^2(theta)(1-sin^2(36°)) + 2sin(theta)cos(theta)sin(36°)cos(36°) + cos^2(theta)(1-cos^2(36°))] + (sin(theta))^2 = (cos(theta))^2
Simplifying further by substituting known values:
[sin^2(theta)(1-[(1+sqrt(5))/4]) + 2sin(theta)cos(theta)sin(36°)cos(36°) + cos^2(theta)(1-[(1-sqrt(5))/4])] + (sin(theta))^2 = (cos(theta))^2
Now we can rearrange the equation to isolate the terms involving theta:
[sin^2(theta)(1-[(1+sqrt(5))/4]) - (cos^2(theta)(1-[(1-sqrt(5))/4]))] + (sin(theta))^2 - (cos(theta))^2 + 2sin(theta)cos(theta)sin(36°)cos(36°) = 0
At this point, we have a quadratic equation with respect to theta. By simplifying and collecting terms, we can solve for theta:
[(sin^2(theta) - cos^2(theta))(1+sqrt(5))/4] + (sin(theta))^2 - (cos(theta))^2 + 2sin(theta)cos(theta)sin(36°)cos(36°) = 0
Using the trigonometric identity for the difference of squares:
[(sin(theta) + cos(theta))(sin(theta) - cos(theta))(1+sqrt(5))/4] + (sin(theta))^2 - (cos(theta))^2 + 2sin(theta)cos(theta)sin(36°)cos(36°) = 0
Now we have two possible cases:
Case 1: sin(theta) + cos(theta) = 0
In this case, we can solve for theta by setting sin(theta) = -cos(theta) and solving the resulting equation.
Case 2: sin(theta) - cos(theta) = 0
In this case, we can solve for theta by setting sin(theta) = cos(theta) and solving the resulting equation.
Please note that the above approach may involve complex calculations and may not yield a simple or exact value for theta.
sin(θ+36) = cosθ
sinθ cos36 + cosθ sin36 = cosθ
sinθ cos36 = cosθ(1-sin36)
That's exact, but you can approximate
sin 36 = 0.6
cos 36 = 0.8
You can change sin to cos bu squaring and remembering that sin^2 + cos^2 = 1
Your School SUBJECT is probably MATH.
or, you can proceed by noting that you have
tanθ = (1-sin36)/cos36 = 0.509