if sin theta equals 3/7, what is tan theta?
sinA = 3/7 = Y/r.
x^2 + y^2 = r^2
x^2 + 3^2 = 7^2.
x^2 = 7^2 - 3^2 = 40.
X = sqrt(4*10) = 2*sqrt10.
tanA=Y/X = 3 / 2*sqrt10 = 3*sqrt10/20.
NOTE: The radical was removed from the
denominator by multiplying numerator and denominator by sqrt10.
sin theta = o/h
that leaves adjacent side open
a^2 + b^2 = c^2
3^2 + b^2 = 7^2
9 + b^2 = 49
b^2 = 40
b = srt 40 adj side now equals sqrt 40
tan theta = o/a = 3/sqrt 40 =
3 * sqrt 40/40
To find the value of tan(theta), given that sin(theta) equals 3/7, we can use the following trigonometric identity:
tan(theta) = sin(theta) / cos(theta)
To calculate cos(theta), we can use the Pythagorean identity:
sin^2(theta) + cos^2(theta) = 1
Substituting the value of sin(theta) into the equation, we have:
(3/7)^2 + cos^2(theta) = 1
Simplifying:
9/49 + cos^2(theta) = 1
Now, let's solve for cos(theta):
cos^2(theta) = 1 - 9/49
cos^2(theta) = 40/49
Taking the square root of both sides:
cos(theta) = sqrt(40/49)
cos(theta) = sqrt(40) / sqrt(49)
cos(theta) = 2sqrt(10) / 7
Now, substituting the values of sin(theta) and cos(theta) into the formula for tan(theta):
tan(theta) = (3/7) / (2sqrt(10) / 7)
tan(theta) = (3/7) * (7 / 2sqrt(10))
tan(theta) = 3 / (2sqrt(10))
tan(theta) = (3/2) * (1/sqrt(10))
tan(theta) = (3/2) * (sqrt(10) / 10)
tan(theta) = 3sqrt(10) / 20
Therefore, the value of tan(theta) when sin(theta) equals 3/7 is 3sqrt(10) / 20.
To find the value of tan(theta), we can use the relationship between the sine and tangent functions. Here's how to do it:
Step 1: Recall the basic trigonometric identities:
- sin(theta) = opposite/hypotenuse
- tan(theta) = opposite/adjacent
Step 2: Since sin(theta) = 3/7, we can assume a right-angled triangle where the opposite side is 3 and the hypotenuse is 7. Let's label the adjacent side as 'a' for now.
Step 3: To find the length of the adjacent side, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
By substituting the known values, we get:
7^2 = a^2 + 3^2
49 = a^2 + 9
a^2 = 49 - 9
a^2 = 40
a = √40 = 2√10 (taking the positive square root as length cannot be negative)
Step 4: Now that we have the values for the opposite side (3) and the adjacent side (2√10), we can substitute them into the tan(theta) formula:
tan(theta) = opposite/adjacent
tan(theta) = 3/(2√10)
Therefore, the value of tan(theta) is 3/(2√10).