if 5sin theta + 12cos theta is equal to 13 find the value of tan theta
To find the value of tan(theta), we need to rearrange the given equation to isolate the trigonometric function. Let's start by squaring both sides of the equation:
(5sin(theta) + 12cos(theta))^2 = 13^2
25sin^2(theta) + 120sin(theta)cos(theta) + 144cos^2(theta) = 169
Using the Pythagorean identity sin^2(theta) + cos^2(theta) = 1, we can substitute that into the equation:
25(1 - cos^2(theta)) + 120sin(theta)cos(theta) + 144cos^2(theta) = 169
25 - 25cos^2(theta) + 120sin(theta)cos(theta) + 144cos^2(theta) = 169
Combine like terms:
24cos^2(theta) + 120sin(theta)cos(theta) - 144 = 0
Divide each term by 24 to simplify:
cos^2(theta) + 5sin(theta)cos(theta) - 6 = 0
Now we have a quadratic equation in terms of cos(theta). Let's solve it using factoring:
(cos(theta) + 6)(cos(theta) - 1) = 0
Setting each factor equal to zero:
cos(theta) + 6 = 0 or cos(theta) - 1 = 0
Solving for cos(theta):
cos(theta) = -6 or cos(theta) = 1
Since the range of the cosine function is -1 to 1, the value of cos(theta) cannot be -6. Therefore, we have:
cos(theta) = 1
Now we can use the identity tan(theta) = sin(theta) / cos(theta) to find the value of tan(theta):
tan(theta) = sin(theta) / cos(theta) = sin(theta) / 1 = sin(theta)
Hence, the value of tan(theta) is equal to sin(theta).
To find the value of tan theta, we need to utilize the relationship between sin and cos. Recall that tan theta is equal to sin theta divided by cos theta.
Given that 5sin theta + 12cos theta = 13, we can rearrange the equation to isolate sin theta:
5sin theta = 13 - 12cos theta
Next, divide both sides of the equation by 5:
sin theta = (13 - 12cos theta) / 5
Now, we can use the Pythagorean identity to solve for cos theta. The Pythagorean identity states that sin^2 theta + cos^2 theta = 1.
Substituting sin theta with the given value, we get:
((13 - 12cos theta) / 5)^2 + cos^2 theta = 1
Expanding and simplifying the equation, we have:
(169 - 312cos theta + 144cos^2 theta) / 25 + cos^2 theta = 1
Multiplying both sides of the equation by 25 to eliminate the denominator, we obtain:
169 - 312cos theta + 144cos^2 theta + 25cos^2 theta = 25
Combining like terms:
169 + 169cos^2 theta - 312cos theta = 25
Rearranging the equation:
169cos^2 theta - 312cos theta + 144 = 0
Now we can solve this quadratic equation to find the value of cos theta. Once we have the value of cos theta, we can substitute it back into the equation sin theta = (13 - 12cos theta) / 5 to find the value of sin theta. Finally, we can calculate tan theta by dividing sin theta by cos theta.
Thank you sooo much....it helps
Since (5,12,13) is a Pythagorean triplet, tan(θ)=5/12.
Mathematically, we solve it by:
Divide by the right hand side:
(5/13)sin(θ)+(12/13)cos(θ)=1
Since (5/13)²+(12/13)²=1, we can put
5/13=sin(φ)
12/13=cos(φ)
so
sin(φ)sin(θ)+cos(φ)cos(&theta)=1
cos(θ-φ)=1
θ-φ=0
θ=φ=asin(5/13).