If θ is an acute angle and tan(θ)=5/12, then sin(θ)=??
To find the value of sin(θ), we can use the Pythagorean identity involving the tan(θ).
The Pythagorean identity is: tan^2(θ) + 1 = sec^2(θ)
From the given information, we know that tan(θ) = 5/12.
Substituting this value into the Pythagorean identity, we get:
(5/12)^2 + 1 = sec^2(θ)
Simplifying this equation, we have:
25/144 + 1 = sec^2(θ)
Multiplying both sides by 144 to eliminate the fraction, we get:
25 + 144 = 144 * sec^2(θ)
169 = 144 * sec^2(θ)
Now we can find the value of sec(θ):
sec^2(θ) = 169/144
Taking the square root of both sides, we have:
sec(θ) = √(169/144)
sec(θ) = 13/12
Since sec(θ) is the reciprocal of cos(θ), we can find the value of cos(θ) as:
cos(θ) = 12/13
Finally, to find the value of sin(θ), we can use the Pythagorean identity involving sin(θ) and cos(θ):
sin^2(θ) + cos^2(θ) = 1
Substituting the value of cos(θ) = 12/13 into the equation, we have:
sin^2(θ) + (12/13)^2 = 1
sin^2(θ) + 144/169 = 1
Subtracting 144/169 from both sides, we get:
sin^2(θ) = 169/169 - 144/169
sin^2(θ) = 25/169
Taking the square root of both sides, we have:
sin(θ) = √(25/169)
simplifying this, we get:
sin(θ) = 5/13
Therefore, sin(θ) = 5/13.
To find the value of sin(θ), we can use the Pythagorean identity for right triangles.
Since θ is an acute angle, we can draw a right triangle with θ as one of the acute angles. Let's label the sides of the triangle based on the information given:
Opposite side = 5
Adjacent side = 12
Using the Pythagorean theorem, we can find the length of the hypotenuse (denoted as h):
h^2 = (Opposite side)^2 + (Adjacent side)^2
h^2 = 5^2 + 12^2
h^2 = 25 + 144
h^2 = 169
Taking the square root of both sides, we get:
h = √169
h = 13
Now, we can find the value of sin(θ) by dividing the length of the opposite side by the length of the hypotenuse:
sin(θ) = (Opposite side) / (Hypotenuse)
sin(θ) = 5 / 13
Therefore, sin(θ) = 5/13.
Think of a 5-12-13 triangle.
Draw the triangle and review your basic trig functions.
tan(θ) = y/x
= 5/12
Find r
r^2 = x^2 +y^2
= 12^2 + 5^2
= 144 + 25
r = Square Root (169)
r = 13
sin(θ) = y/r = 5/13