Given The Ordered Pair \[$ P(6, -5) \$\], Determine The Requested \[$\sin, \cos\$\], Or \[$\tan\$\] Of Angle \[$\theta\$\].\[$\sin \theta =\$\]A. \[$-\frac{6\sqrt{61}}{61}\$\]B.
Introduction
In trigonometry, ordered pairs are used to represent points on the Cartesian plane. These pairs consist of an x-coordinate and a y-coordinate, which can be used to calculate various trigonometric functions such as sine, cosine, and tangent. In this article, we will explore how to determine the requested trigonometric function of an angle given an ordered pair.
Understanding the Ordered Pair
The ordered pair given is (6, -5). This means that the x-coordinate is 6 and the y-coordinate is -5. To calculate the trigonometric functions, we need to use these coordinates to find the values of sine, cosine, and tangent.
Calculating Sine
To calculate the sine of an angle, we use the formula:
sin(θ) = y / r
where y is the y-coordinate and r is the distance from the origin to the point.
First, we need to find the distance from the origin to the point (6, -5). We can use the Pythagorean theorem to find this distance:
r = √(x^2 + y^2) = √(6^2 + (-5)^2) = √(36 + 25) = √61
Now that we have the distance, we can calculate the sine of the angle:
sin(θ) = y / r = -5 / √61 = -5 / √61 × (√61 / √61) = -5√61 / 61
Calculating Cosine
To calculate the cosine of an angle, we use the formula:
cos(θ) = x / r
where x is the x-coordinate and r is the distance from the origin to the point.
We already found the distance from the origin to the point (6, -5) in the previous section:
r = √61
Now, we can calculate the cosine of the angle:
cos(θ) = x / r = 6 / √61 = 6 / √61 × (√61 / √61) = 6√61 / 61
Calculating Tangent
To calculate the tangent of an angle, we use the formula:
tan(θ) = y / x
where y is the y-coordinate and x is the x-coordinate.
We already found the values of y and x in the previous sections:
y = -5 x = 6
Now, we can calculate the tangent of the angle:
tan(θ) = y / x = -5 / 6
Conclusion
In this article, we calculated the sine, cosine, and tangent of an angle given the ordered pair (6, -5). We used the formulas for each trigonometric function and the distance from the origin to the point to find the values. The sine of the angle is -5√61 / 61, the cosine of the angle is 6√61 / 61, and the tangent of the angle is -5 / 6.
Calculating Trigonometric Functions from an Ordered Pair: Step-by-Step Guide
Step 1: Understand the Ordered Pair
The ordered pair given is (6, -5). This means that the x-coordinate is 6 and the y-coordinate is -5.
Step 2: Calculate the Distance from the Origin to the Point
To calculate the distance from the origin to the point (6, -5), we use the Pythagorean theorem:
r = √(x^2 + y^2) = √(6^2 + (-5)^2) = √(36 + 25) = √61
Step 3: Calculate the Sine of the Angle
To calculate the sine of the angle, we use the formula:
sin(θ) = y / r
where y is the y-coordinate and r is the distance from the origin to the point.
sin(θ) = -5 / √61 = -5 / √61 × (√61 / √61) = -5√61 / 61
Step 4: Calculate the Cosine of the Angle
To calculate the cosine of the angle, we use the formula:
cos(θ) = x / r
where x is the x-coordinate and r is the distance from the origin to the point.
cos(θ) = 6 / √61 = 6 / √61 × (√61 / √61) = 6√61 / 61
Step 5: Calculate the Tangent of the Angle
To calculate the tangent of the angle, we use the formula:
tan(θ) = y / x
where y is the y-coordinate and x is the x-coordinate.
tan(θ) = -5 / 6
Frequently Asked Questions
Q: What is the ordered pair given in this article?
A: The ordered pair given is (6, -5).
Q: How do we calculate the sine of an angle from an ordered pair?
A: To calculate the sine of an angle from an ordered pair, we use the formula sin(θ) = y / r, where y is the y-coordinate and r is the distance from the origin to the point.
Q: How do we calculate the cosine of an angle from an ordered pair?
A: To calculate the cosine of an angle from an ordered pair, we use the formula cos(θ) = x / r, where x is the x-coordinate and r is the distance from the origin to the point.
Q: How do we calculate the tangent of an angle from an ordered pair?
A: To calculate the tangent of an angle from an ordered pair, we use the formula tan(θ) = y / x, where y is the y-coordinate and x is the x-coordinate.
Conclusion
In this article, we calculated the sine, cosine, and tangent of an angle given the ordered pair (6, -5). We used the formulas for each trigonometric function and the distance from the origin to the point to find the values. The sine of the angle is -5√61 / 61, the cosine of the angle is 6√61 / 61, and the tangent of the angle is -5 / 6.
Introduction
In our previous article, we explored how to calculate the sine, cosine, and tangent of an angle given an ordered pair. We used the formulas for each trigonometric function and the distance from the origin to the point to find the values. In this article, we will answer some frequently asked questions about calculating trigonometric functions from an ordered pair.
Q&A
Q: What is the ordered pair given in this article?
A: The ordered pair given is (6, -5).
Q: How do we calculate the sine of an angle from an ordered pair?
A: To calculate the sine of an angle from an ordered pair, we use the formula sin(θ) = y / r, where y is the y-coordinate and r is the distance from the origin to the point.
Q: How do we calculate the cosine of an angle from an ordered pair?
A: To calculate the cosine of an angle from an ordered pair, we use the formula cos(θ) = x / r, where x is the x-coordinate and r is the distance from the origin to the point.
Q: How do we calculate the tangent of an angle from an ordered pair?
A: To calculate the tangent of an angle from an ordered pair, we use the formula tan(θ) = y / x, where y is the y-coordinate and x is the x-coordinate.
Q: What is the distance from the origin to the point (6, -5)?
A: The distance from the origin to the point (6, -5) is √61.
Q: How do we find the distance from the origin to a point?
A: To find the distance from the origin to a point, we use the Pythagorean theorem:
r = √(x^2 + y^2)
Q: What is the sine of the angle given the ordered pair (6, -5)?
A: The sine of the angle given the ordered pair (6, -5) is -5√61 / 61.
Q: What is the cosine of the angle given the ordered pair (6, -5)?
A: The cosine of the angle given the ordered pair (6, -5) is 6√61 / 61.
Q: What is the tangent of the angle given the ordered pair (6, -5)?
A: The tangent of the angle given the ordered pair (6, -5) is -5 / 6.
Q: Can we use the ordered pair (6, -5) to find the values of sine, cosine, and tangent for any angle?
A: No, the ordered pair (6, -5) is specific to the angle θ, and we can only use it to find the values of sine, cosine, and tangent for that particular angle.
Q: How do we use the ordered pair (6, -5) to find the values of sine, cosine, and tangent?
A: To use the ordered pair (6, -5) to find the values of sine, cosine, and tangent, we need to follow the steps outlined in our previous article:
- Calculate the distance from the origin to the point (6, -5).
- Use the distance to calculate the sine of the angle.
- Use the distance to calculate the cosine of the angle.
- Use the distance to calculate the tangent of the angle.
Conclusion
In this article, we answered some frequently asked questions about calculating trigonometric functions from an ordered pair. We covered topics such as the ordered pair, the distance from the origin to a point, and the formulas for calculating sine, cosine, and tangent. We also provided examples of how to use the ordered pair (6, -5) to find the values of sine, cosine, and tangent for a particular angle.
Calculating Trigonometric Functions from an Ordered Pair: Step-by-Step Guide
Step 1: Understand the Ordered Pair
The ordered pair given is (6, -5). This means that the x-coordinate is 6 and the y-coordinate is -5.
Step 2: Calculate the Distance from the Origin to the Point
To calculate the distance from the origin to the point (6, -5), we use the Pythagorean theorem:
r = √(x^2 + y^2) = √(6^2 + (-5)^2) = √(36 + 25) = √61
Step 3: Calculate the Sine of the Angle
To calculate the sine of the angle, we use the formula:
sin(θ) = y / r
where y is the y-coordinate and r is the distance from the origin to the point.
sin(θ) = -5 / √61 = -5 / √61 × (√61 / √61) = -5√61 / 61
Step 4: Calculate the Cosine of the Angle
To calculate the cosine of the angle, we use the formula:
cos(θ) = x / r
where x is the x-coordinate and r is the distance from the origin to the point.
cos(θ) = 6 / √61 = 6 / √61 × (√61 / √61) = 6√61 / 61
Step 5: Calculate the Tangent of the Angle
To calculate the tangent of the angle, we use the formula:
tan(θ) = y / x
where y is the y-coordinate and x is the x-coordinate.
tan(θ) = -5 / 6
Frequently Asked Questions
Q: What is the ordered pair given in this article?
A: The ordered pair given is (6, -5).
Q: How do we calculate the sine of an angle from an ordered pair?
A: To calculate the sine of an angle from an ordered pair, we use the formula sin(θ) = y / r, where y is the y-coordinate and r is the distance from the origin to the point.
Q: How do we calculate the cosine of an angle from an ordered pair?
A: To calculate the cosine of an angle from an ordered pair, we use the formula cos(θ) = x / r, where x is the x-coordinate and r is the distance from the origin to the point.
Q: How do we calculate the tangent of an angle from an ordered pair?
A: To calculate the tangent of an angle from an ordered pair, we use the formula tan(θ) = y / x, where y is the y-coordinate and x is the x-coordinate.
Conclusion
In this article, we provided a step-by-step guide on how to calculate the sine, cosine, and tangent of an angle given an ordered pair. We also answered some frequently asked questions about calculating trigonometric functions from an ordered pair.