Given That $\tan \theta = \frac{8}{15}$, And That $180^{\circ} \ \textless \ \theta \ \textless \ 270^{\circ}$, Find The Value Of $\sin \theta$ And $\cos \theta$ Without Using Tables.

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Solving Trigonometric Equations: Finding Sin and Cos Values

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

We are given that tan⁡θ=815\tan \theta = \frac{8}{15}, and that 180∘ \textless θ \textless 270∘180^{\circ} \ \textless \ \theta \ \textless \ 270^{\circ}. This means that the angle θ\theta is in the third quadrant of the unit circle. Our goal is to find the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta without using tables.

Before we proceed, let's recall some essential trigonometric identities:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
  • sin⁡θ=oppositehypotenuse\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}
  • cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}

We can use the Pythagorean identity to express sin⁡2θ\sin^2 \theta and cos⁡2θ\cos^2 \theta in terms of tan⁡θ\tan \theta. Since tan⁡θ=815\tan \theta = \frac{8}{15}, we can write:

tan⁡2θ=(815)2=64225\tan^2 \theta = \left(\frac{8}{15}\right)^2 = \frac{64}{225}

Using the Pythagorean identity, we can express sin⁡2θ\sin^2 \theta and cos⁡2θ\cos^2 \theta as:

sin⁡2θ=tan⁡2θ1+tan⁡2θ=642251+64225=64289\sin^2 \theta = \frac{\tan^2 \theta}{1 + \tan^2 \theta} = \frac{\frac{64}{225}}{1 + \frac{64}{225}} = \frac{64}{289}

cos⁡2θ=11+tan⁡2θ=11+64225=225289\cos^2 \theta = \frac{1}{1 + \tan^2 \theta} = \frac{1}{1 + \frac{64}{225}} = \frac{225}{289}

Now that we have expressed sin⁡2θ\sin^2 \theta and cos⁡2θ\cos^2 \theta in terms of tan⁡θ\tan \theta, we can find the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta. Since θ\theta is in the third quadrant, both sin⁡θ\sin \theta and cos⁡θ\cos \theta are negative.

sin⁡θ=−sin⁡2θ=−64289=−817\sin \theta = -\sqrt{\sin^2 \theta} = -\sqrt{\frac{64}{289}} = -\frac{8}{17}

cos⁡θ=−cos⁡2θ=−225289=−1517\cos \theta = -\sqrt{\cos^2 \theta} = -\sqrt{\frac{225}{289}} = -\frac{15}{17}

In this article, we have shown how to find the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta given the tangent value and the quadrant of the angle. We used the Pythagorean identity to express sin⁡2θ\sin^2 \theta and cos⁡2θ\cos^2 \theta in terms of tan⁡θ\tan \theta, and then found the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta by taking the square root of the expressions. This method can be applied to solve trigonometric equations in various quadrants.

Trigonometry has numerous applications in various fields, including:

  • Physics: Trigonometry is used to describe the motion of objects in terms of position, velocity, and acceleration.
  • Engineering: Trigonometry is used to design and analyze electrical circuits, mechanical systems, and other engineering applications.
  • Navigation: Trigonometry is used in navigation systems, such as GPS, to determine the position and velocity of objects.
  • Computer Science: Trigonometry is used in computer graphics, game development, and other applications that require 3D modeling and animation.

In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. By understanding the relationships between the sides and angles of triangles, we can solve trigonometric equations and find the values of sine and cosine functions. This article has shown how to find the values of sin⁡θ\sin \theta and cos⁡θ\cos \theta given the tangent value and the quadrant of the angle. We hope that this article has provided a clear and concise explanation of the subject, and that it will be helpful to students and professionals alike.
Trigonometry Q&A: Frequently Asked Questions

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

A: The sine, cosine, and tangent functions are three fundamental trigonometric functions that describe the relationships between the sides and angles of triangles. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse, the cosine function is defined as the ratio of the length of the adjacent side to the length of the hypotenuse, and the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side.

A: One way to remember the order of the sine, cosine, and tangent functions is to use the mnemonic "SOH-CAH-TOA". This stands for:

  • Sine = Opposite over Hypotenuse (SOH)
  • Cosine = Adjacent over Hypotenuse (CAH)
  • Tangent = Opposite over Adjacent (TOA)

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

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

A: There are several ways to find the value of the sine, cosine, and tangent functions for a given angle. One way is to use a calculator or a trigonometric table. Another way is to use the unit circle, which is a circle with a radius of 1 centered at the origin of the coordinate plane. The sine, cosine, and tangent functions can be defined in terms of the coordinates of points on the unit circle.

A: The Pythagorean identity is a fundamental trigonometric identity that states:

sin^2(x) + cos^2(x) = 1

This identity can be used to find the value of the sine or cosine function for a given angle, as long as the value of the other function is known.

A: To use the Pythagorean identity to find the value of the sine or cosine function, you can start by squaring the value of the other function. Then, you can add the squared value to 1 and take the square root of the result. This will give you the value of the sine or cosine function.

A: Some common trigonometric identities include:

  • sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
  • cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
  • tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))

These identities can be used to simplify trigonometric expressions and to find the value of the sine, cosine, and tangent functions for a given angle.

A: Trigonometry has numerous applications in various fields, including physics, engineering, and navigation. Some examples of real-world applications of trigonometry include:

  • Calculating the trajectory of a projectile
  • Designing and analyzing electrical circuits
  • Determining the position and velocity of objects in navigation systems
  • Creating 3D models and animations in computer graphics

In conclusion, trigonometry is a fundamental subject that has numerous applications in various fields. By understanding the relationships between the sides and angles of triangles, we can solve trigonometric equations and find the values of sine and cosine functions. This article has answered some frequently asked questions about trigonometry, covering topics such as sine, cosine, and tangent functions, as well as their applications. We hope that this article has provided a clear and concise explanation of the subject, and that it will be helpful to students and professionals alike.