Evaluate using the Pythagorean indentities
Find sinθ and cosθ if tanθ =1/6 and sinθ >0
Help please ! Thank you !
draw the right triangle
opposite 1
adjacent 6
hypotenuse= sqrt37 ie sqrt(6^2+1^2)
sinTheta=1/sqrt37
costheta=6/sqrt37
To solve this problem, we will use the Pythagorean identities, which are:
1. sin²θ + cos²θ = 1
2. tan²θ + 1 = sec²θ
Given that tanθ = 1/6 and sinθ > 0, we can find the values of sinθ and cosθ.
Step 1: Use the given tangent value to find the values of sinθ and cosθ.
Since tanθ = 1/6, we can set up the equation:
tanθ = sinθ / cosθ
Substituting the given value, we get:
1/6 = sinθ / cosθ
Step 2: Simplify the equation.
To simplify the equation, we can multiply both sides by cosθ:
(1/6) * cosθ = sinθ
Now, we have an equation that relates sinθ and cosθ.
Step 3: Use the Pythagorean identity sin²θ + cos²θ = 1 to solve for the remaining unknown value.
Substitute the expression for sinθ from the previous step into the Pythagorean identity:
(sinθ)² + cos²θ = 1
Replacing sinθ with (1/6) * cosθ in the equation:
((1/6) * cosθ)² + cos²θ = 1
Step 4: Solve for cosθ.
To solve for cosθ, we need to simplify the equation. Expanding and rearranging terms, we get:
(1/36) * cos²θ + cos²θ = 1
Multiply through by 36 to eliminate the fractions:
cos²θ + 36 * cos²θ = 36
Combine like terms:
37 * cos²θ = 36
Divide by 37 to isolate cos²θ:
cos²θ = 36 / 37
Take the square root of both sides to solve for cosθ:
cosθ = ± √(36 / 37)
Since sinθ > 0, we know that sinθ is positive. Therefore, we can choose the positive root for cosθ:
cosθ = √(36 / 37)
Step 5: Use the value of cosθ to find sinθ.
Now that we have the value of cosθ, we can substitute it back into the equation we derived in Step 2:
(1/6) * cosθ = sinθ
Substituting the value of cosθ, we get:
(1/6) * √(36 / 37) = sinθ
Finally, we can simplify this expression to find the value of sinθ.
Evaluating the expression, we get:
sinθ ≈ 0.1005
cosθ ≈ 0.9949
Therefore, sinθ ≈ 0.1005 and cosθ ≈ 0.9949 when tanθ = 1/6 and sinθ > 0.