Given The Ordered Pair \[$ P(-5, -3) \$\], Determine The Requested \[$\sin \theta\$\], \[$\cos \theta\$\], Or \[$\tan \theta\$\] Of Angle \[$\theta\$\].\[$\sin \theta =\$\]A. \[$-\frac{3
Trigonometric Ratios: Finding Sine, Cosine, and Tangent of an Angle
In trigonometry, the sine, cosine, and tangent of an angle are defined as the ratios of the lengths of the sides of a right triangle. Given the ordered pair { P(-5, -3) $}$, we are tasked with determining the requested {\sin \theta$}$, {\cos \theta$}$, or {\tan \theta$}$ of angle {\theta$}$. To accomplish this, we need to understand the relationships between the coordinates of a point and the trigonometric ratios.
The sine, cosine, and tangent of an angle are defined as follows:
- Sine: {\sin \theta = \frac{opposite}{hypotenuse}$]
- Cosine: [$\cos \theta = \frac{adjacent}{hypotenuse}$]
- Tangent: [$\tan \theta = \frac{opposite}{adjacent}$]
In a right triangle, the opposite side is the side opposite the angle, the adjacent side is the side adjacent to the angle, and the hypotenuse is the side opposite the right angle.
Given the ordered pair [$ P(-5, -3) $}$, we can use the coordinates to find the trigonometric ratios. The x-coordinate represents the adjacent side, and the y-coordinate represents the opposite side.
Finding Sine
To find the sine of the angle, we need to divide the opposite side by the hypotenuse. However, we are not given the hypotenuse. We can use the Pythagorean theorem to find the hypotenuse:
{c^2 = a^2 + b^2$]
where [a$] is the adjacent side, and [$b$] is the opposite side.
Plugging in the values, we get:
[$c^2 = (-5)^2 + (-3)^2$]
[$c^2 = 25 + 9$]
[$c^2 = 34$]
Taking the square root of both sides, we get:
[$c = \sqrt{34}$]
Now that we have the hypotenuse, we can find the sine of the angle:
[$\sin \theta = \frac{opposite}{hypotenuse} = \frac{-3}{\sqrt{34}}$]
Finding Cosine
To find the cosine of the angle, we need to divide the adjacent side by the hypotenuse. We can use the coordinates to find the adjacent side:
[$\cos \theta = \frac{adjacent}{hypotenuse} = \frac{-5}{\sqrt{34}}$]
Finding Tangent
To find the tangent of the angle, we need to divide the opposite side by the adjacent side. We can use the coordinates to find the opposite and adjacent sides:
[$\tan \theta = \frac{opposite}{adjacent} = \frac{-3}{-5} = \frac{3}{5}$]
In this article, we have discussed how to find the sine, cosine, and tangent of an angle given the ordered pair [$ P(-5, -3) $}$. We used the Pythagorean theorem to find the hypotenuse and then used the coordinates to find the trigonometric ratios. The sine of the angle is {-\frac{3}{\sqrt{34}}$, the cosine of the angle is [\frac{3}{5}$.
- Navigation: Trigonometric ratios are used in navigation to determine the direction and distance of an object.
- Physics: Trigonometric ratios are used in physics to describe the motion of objects.
- Engineering: Trigonometric ratios are used in engineering to design and build structures.
- Not using the correct coordinates: Make sure to use the correct coordinates to find the trigonometric ratios.
- Not using the Pythagorean theorem: Make sure to use the Pythagorean theorem to find the hypotenuse.
- Not simplifying the expressions: Make sure to simplify the expressions to get the final answer.
- Use the unit circle: The unit circle is a useful tool for finding trigonometric ratios.
- Use the Pythagorean theorem: The Pythagorean theorem is a useful tool for finding the hypotenuse.
- Simplify the expressions: Simplifying the expressions will make it easier to get the final answer.
Trigonometric Ratios: Q&A
In our previous article, we discussed how to find the sine, cosine, and tangent of an angle given the ordered pair [$ P(-5, -3) $}$. In this article, we will answer some frequently asked questions about trigonometric ratios.
Q: What is the difference between sine, cosine, and tangent?
A: The sine, cosine, and tangent of an angle are defined as the ratios of the lengths of the sides of a right triangle. The sine is the ratio of the opposite side to the hypotenuse, the cosine is the ratio of the adjacent side to the hypotenuse, and the tangent is the ratio of the opposite side to the adjacent side.
Q: How do I find the sine, cosine, and tangent of an angle?
A: To find the sine, cosine, and tangent of an angle, you need to know the lengths of the sides of the right triangle. You can use the Pythagorean theorem to find the hypotenuse, and then use the coordinates to find the trigonometric ratios.
Q: What is the Pythagorean theorem?
A: The Pythagorean theorem is a mathematical formula that describes the relationship between the lengths of the sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Q: How do I use the Pythagorean theorem to find the hypotenuse?
A: To use the Pythagorean theorem to find the hypotenuse, you need to know the lengths of the other two sides. You can then plug these values into the formula and solve for the hypotenuse.
Q: What is the unit circle?
A: The unit circle is a circle with a radius of 1 that is centered at the origin of a coordinate plane. It is a useful tool for finding trigonometric ratios.
Q: How do I use the unit circle to find trigonometric ratios?
A: To use the unit circle to find trigonometric ratios, you need to know the coordinates of a point on the circle. You can then use these coordinates to find the trigonometric ratios.
Q: What are some common mistakes to avoid when finding trigonometric ratios?
A: Some common mistakes to avoid when finding trigonometric ratios include not using the correct coordinates, not using the Pythagorean theorem, and not simplifying the expressions.
Q: What are some tips and tricks for finding trigonometric ratios?
A: Some tips and tricks for finding trigonometric ratios include using the unit circle, using the Pythagorean theorem, and simplifying the expressions.
Q: How do I apply trigonometric ratios in real-life situations?
A: Trigonometric ratios are used in a variety of real-life situations, including navigation, physics, and engineering. They can be used to describe the motion of objects, determine the direction and distance of an object, and design and build structures.
In this article, we have answered some frequently asked questions about trigonometric ratios. We have discussed the difference between sine, cosine, and tangent, how to find the trigonometric ratios, and some common mistakes to avoid. We have also provided some tips and tricks for finding trigonometric ratios and applying them in real-life situations.
- Navigation: Trigonometric ratios are used in navigation to determine the direction and distance of an object.
- Physics: Trigonometric ratios are used in physics to describe the motion of objects.
- Engineering: Trigonometric ratios are used in engineering to design and build structures.
- Not using the correct coordinates: Make sure to use the correct coordinates to find the trigonometric ratios.
- Not using the Pythagorean theorem: Make sure to use the Pythagorean theorem to find the hypotenuse.
- Not simplifying the expressions: Make sure to simplify the expressions to get the final answer.
- Use the unit circle: The unit circle is a useful tool for finding trigonometric ratios.
- Use the Pythagorean theorem: The Pythagorean theorem is a useful tool for finding the hypotenuse.
- Simplify the expressions: Simplifying the expressions will make it easier to get the final answer.