You can find the sine (sin) of an angle if you know the tangent (tan) by using trigonometric identities and understanding the relationships between sine, cosine, and tangent. Here's how:
1. Understanding the Relationship
- Recall the fundamental trigonometric definitions:
- sin(θ) = Opposite / Hypotenuse
- cos(θ) = Adjacent / Hypotenuse
- tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)
2. Using the Pythagorean Identity
- We know that sin2(θ) + cos2(θ) = 1. This is the Pythagorean identity.
- We also know tan(θ) = sin(θ) / cos(θ). Therefore, cos(θ) = sin(θ) / tan(θ).
3. Substitution and Solving
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Substitute cos(θ) = sin(θ) / tan(θ) into the Pythagorean identity:
sin2(θ) + (sin(θ) / tan(θ))2 = 1
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Simplify the equation:
sin2(θ) + sin2(θ) / tan2(θ) = 1
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Factor out sin2(θ):
sin2(θ) * (1 + 1 / tan2(θ)) = 1
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Isolate sin2(θ):
sin2(θ) = 1 / (1 + 1 / tan2(θ))
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Simplify further:
sin2(θ) = tan2(θ) / (tan2(θ) + 1)
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Solve for sin(θ):
sin(θ) = ± √(tan2(θ) / (tan2(θ) + 1))
sin(θ) = ± tan(θ) / √(tan2(θ) + 1)
4. Determining the Sign
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The ± sign indicates that there are two possible values for sin(θ), one positive and one negative. You need to determine the correct sign based on the quadrant in which the angle θ lies. Consider the following:
- Quadrant I (0° < θ < 90°): sin(θ) > 0 and tan(θ) > 0
- Quadrant II (90° < θ < 180°): sin(θ) > 0 and tan(θ) < 0
- Quadrant III (180° < θ < 270°): sin(θ) < 0 and tan(θ) > 0
- Quadrant IV (270° < θ < 360°): sin(θ) < 0 and tan(θ) < 0
Example:
If tan(θ) = 1 and θ is in Quadrant I, then:
sin(θ) = + 1 / √(12 + 1) = 1 / √2 = √2 / 2
In summary, to find sin(θ) from tan(θ), use the formula sin(θ) = ± tan(θ) / √(tan2(θ) + 1), and determine the correct sign based on the quadrant of the angle.
