If $\tan \theta = \frac{3}{4}$, Find $\sin \theta$.A. $\sin \theta = \frac{1}{2}$ B. $\sin \theta = \frac{3}{5}$ C. $\sin \theta = 2$ D. $\sin \theta = \frac{8}{5}$ Please Select The Best Answer

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Solving Trigonometric Equations: Finding sin⁡θ\sin \theta Given tan⁡θ\tan \theta

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on solving trigonometric equations, specifically finding the sine of an angle given the tangent of that angle.

Understanding Trigonometric Ratios

Before we dive into solving the equation, let's briefly review the trigonometric ratios. The three basic trigonometric ratios are:

  • Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Given Information

We are given that tan⁡θ=34\tan \theta = \frac{3}{4}. This means that the ratio of the length of the side opposite the angle θ\theta to the length of the side adjacent to the angle θ\theta is 34\frac{3}{4}.

Using the Pythagorean Identity

To find the sine of the angle, we can use the Pythagorean identity:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

We can also use the fact that tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}. Since we are given that tan⁡θ=34\tan \theta = \frac{3}{4}, we can write:

sin⁡θcos⁡θ=34\frac{\sin \theta}{\cos \theta} = \frac{3}{4}

Finding sin⁡θ\sin \theta

To find sin⁡θ\sin \theta, we can use the Pythagorean identity and the fact that tan⁡θ=34\tan \theta = \frac{3}{4}. Let's assume that the length of the side adjacent to the angle θ\theta is 4x4x and the length of the side opposite the angle θ\theta is 3x3x. Then, the length of the hypotenuse is (4x)2+(3x)2=5x\sqrt{(4x)^2 + (3x)^2} = 5x.

Using the Pythagorean identity, we can write:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

(3x5x)2+(4x5x)2=1\left(\frac{3x}{5x}\right)^2 + \left(\frac{4x}{5x}\right)^2 = 1

9x225x2+16x225x2=1\frac{9x^2}{25x^2} + \frac{16x^2}{25x^2} = 1

25x225x2=1\frac{25x^2}{25x^2} = 1

1=11 = 1

This equation is true for all values of xx. Therefore, we can conclude that:

sin⁡θ=3x5x\sin \theta = \frac{3x}{5x}

sin⁡θ=35\sin \theta = \frac{3}{5}

In this article, we solved a trigonometric equation to find the sine of an angle given the tangent of that angle. We used the Pythagorean identity and the fact that tan⁡θ=34\tan \theta = \frac{3}{4} to find the value of sin⁡θ\sin \theta. The final answer is sin⁡θ=35\sin \theta = \frac{3}{5}.

The correct answer is:

  • B. sin⁡θ=35\sin \theta = \frac{3}{5}
  • [1] "Trigonometry" by Michael Corral, 2018.
  • [2] "Precalculus" by James Stewart, 2018.
  • [3] "Trigonometry for Dummies" by Mary Jane Sterling, 2019.
    Trigonometry Q&A: Sine, Cosine, and Tangent

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject in mathematics and has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will answer some frequently asked questions about trigonometry, specifically about sine, cosine, and tangent.

Q: What is the difference between sine, cosine, and tangent?

A: The three basic trigonometric ratios are:

  • Sine (sin): The ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • Cosine (cos): The ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
  • Tangent (tan): The ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Q: How do I remember the sine, cosine, and tangent values?

A: One way to remember the sine, cosine, and tangent values is to use the following mnemonic device:

  • SOH-CAH-TOA: Sine = Opposite over Hypotenuse, Cosine = Adjacent over Hypotenuse, Tangent = Opposite over Adjacent.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

This identity is used to find the values of sine and cosine.

Q: How do I find the value of sine given the value of tangent?

A: To find the value of sine given the value of tangent, you can use the following formula:

sin⁡θ=tan⁡θ1+tan⁡2θ\sin \theta = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}

Q: How do I find the value of cosine given the value of tangent?

A: To find the value of cosine given the value of tangent, you can use the following formula:

cos⁡θ=11+tan⁡2θ\cos \theta = \frac{1}{\sqrt{1 + \tan^2 \theta}}

Q: What is the range of sine, cosine, and tangent?

A: The range of sine, cosine, and tangent is:

  • Sine: [-1, 1]
  • Cosine: [-1, 1]
  • Tangent: [-∞, ∞]

Q: What is the domain of sine, cosine, and tangent?

A: The domain of sine, cosine, and tangent is:

  • Sine: All real numbers
  • Cosine: All real numbers
  • Tangent: All real numbers except 0

In this article, we answered some frequently asked questions about trigonometry, specifically about sine, cosine, and tangent. We covered the differences between sine, cosine, and tangent, how to remember their values, the Pythagorean identity, and how to find their values given the values of other trigonometric ratios.

  • [1] "Trigonometry" by Michael Corral, 2018.
  • [2] "Precalculus" by James Stewart, 2018.
  • [3] "Trigonometry for Dummies" by Mary Jane Sterling, 2019.
  • Q: What is the difference between sine, cosine, and tangent? A: The three basic trigonometric ratios are sine, cosine, and tangent.
  • Q: How do I remember the sine, cosine, and tangent values? A: One way to remember the sine, cosine, and tangent values is to use the mnemonic device SOH-CAH-TOA.
  • Q: What is the Pythagorean identity? A: The Pythagorean identity is sin^2 θ + cos^2 θ = 1.
  • Q: How do I find the value of sine given the value of tangent? A: To find the value of sine given the value of tangent, you can use the formula sin θ = tan θ / √(1 + tan^2 θ).
  • Q: How do I find the value of cosine given the value of tangent? A: To find the value of cosine given the value of tangent, you can use the formula cos θ = 1 / √(1 + tan^2 θ).
  • Q: What is the range of sine, cosine, and tangent? A: The range of sine, cosine, and tangent is [-1, 1] for sine and cosine, and [-∞, ∞] for tangent.
  • Q: What is the domain of sine, cosine, and tangent? A: The domain of sine, cosine, and tangent is all real numbers for sine and cosine, and all real numbers except 0 for tangent.