if sec theta - tan theta ÷sec theta +tan theta = 1÷4 ,find sin theta
I read that as:
(secØ - tanØ)/(secØ + tanØ) = 1/4
4secØ - 4tanØ = secØ + tanØ
3secØ = 5 tanØ
3/cosØ = 5sinØ/cosØ
5sinØcosØ = 3cosØ
5sinØcosØ - 3cosØ = 0
cosØ(5sinØ - 3) = 0
cosØ = 0 , not possible since tanØ would be undefined
sinØ = 3/5
I could have never answered this question without grouping symbols. Reiny is blessed.
To find the value of sin(theta), we can use the relationship between sin, cos, and tan.
Here's how to solve the equation step by step:
1. Rewrite the given equation:
sec(theta) - tan(theta) / sec(theta) + tan(theta) = 1/4
2. Use the definitions of sec(theta) and tan(theta):
1/cos(theta) - sin(theta)/cos(theta) / 1/cos(theta) + sin(theta)/cos(theta) = 1/4
3. Combine the fractions by finding the common denominator:
[1 - sin(theta)] / cos(theta) / [1 + sin(theta)] / cos(theta) = 1/4
4. Invert and multiply:
[1 - sin(theta)] / cos(theta) * cos(theta) / [1 + sin(theta)] = 1/4
Simplify:
(1 - sin(theta)) / (1 + sin(theta)) = 1/4
5. Cross-multiply:
4(1 - sin(theta)) = 1 + sin(theta)
Simplify:
4 - 4sin(theta) = 1 + sin(theta)
6. Move all terms involving sin(theta) to one side of the equation:
4sin(theta) + sin(theta) = 4 - 1
Simplify:
5sin(theta) = 3
7. Divide both sides by 5 to solve for sin(theta):
sin(theta) = 3/5
Therefore, sin(theta) = 3/5.
To find sin theta, we need to manipulate the given equation to get an expression involving sin theta.
Step 1: Consider the given equation:
sec theta - tan theta ÷ sec theta + tan theta = 1 ÷ 4
Step 2: Let's simplify the left-hand side of the equation by multiplying both the numerator and the denominator by (sec theta + tan theta):
(sec theta - tan theta) * (sec theta + tan theta) ÷ (sec theta + tan theta) = 1 ÷ 4
Step 3: Simplify further:
sec^2(theta) - tan^2(theta) = 1 ÷ 4
Step 4: Recall the trigonometric identity: sec^2(theta) - tan^2(theta) = 1
Replacing this identity in the equation, we get:
1 - 1 = 1 ÷ 4
Step 5: Simplify further:
0 = 1 ÷ 4
Step 6: We've reached a contradiction. Therefore, there is no solution for theta that satisfies the given equation.
Hence, we cannot determine the value of sin theta based on the given information.