Draft A Trigonometric Circle And Fill In The Following With The Equivalent Angle For The Related Trigonometric Ratio In The First Quadrant And The Appropriate Algebraic Sign.sin(105°) = Sin( Cos(232°) = Cos(
Introduction
A trigonometric circle is a fundamental concept in mathematics that helps us understand the relationships between angles and trigonometric ratios. In this article, we will explore the trigonometric circle and fill in the missing angles for the given trigonometric ratios in the first quadrant.
What is a Trigonometric Circle?
A trigonometric circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The circle is divided into four quadrants, each representing a different range of angles. The trigonometric circle is used to visualize the relationships between angles and trigonometric ratios, such as sine, cosine, and tangent.
The First Quadrant
The first quadrant of the trigonometric circle is the upper right quadrant, where both x and y coordinates are positive. In this quadrant, the angles range from 0° to 90°. The trigonometric ratios in the first quadrant are positive, and the values of sine, cosine, and tangent are as follows:
- Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Filling in the Missing Angles
Now that we have a basic understanding of the trigonometric circle and the first quadrant, let's fill in the missing angles for the given trigonometric ratios.
sin(105°) = sin(____°)
To find the equivalent angle for sin(105°), we need to use the fact that the sine function is periodic with a period of 360°. This means that sin(105°) is equal to sin(105° + 360°) or sin(465°). However, we want to find an angle in the first quadrant, so we need to subtract 360° from 465°, which gives us 105°. Therefore, sin(105°) = sin(105°).
cos(232°) = cos(____°)
To find the equivalent angle for cos(232°), we need to use the fact that the cosine function is periodic with a period of 360°. This means that cos(232°) is equal to cos(232° + 360°) or cos(592°). However, we want to find an angle in the first quadrant, so we need to subtract 360° from 592°, which gives us 232°. Since 232° is in the third quadrant, we need to find an equivalent angle in the first quadrant. We can do this by subtracting 180° from 232°, which gives us 52°. Therefore, cos(232°) = cos(52°).
cos(____°) = cos(232°)
Since we already found that cos(232°) = cos(52°), we can conclude that cos(____°) = cos(52°).
Conclusion
In this article, we explored the trigonometric circle and filled in the missing angles for the given trigonometric ratios in the first quadrant. We used the periodic properties of the sine and cosine functions to find the equivalent angles in the first quadrant. By understanding the relationships between angles and trigonometric ratios, we can better visualize and solve problems involving trigonometry.
Key Takeaways
- A trigonometric circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
- The first quadrant of the trigonometric circle is the upper right quadrant, where both x and y coordinates are positive.
- The trigonometric ratios in the first quadrant are positive, and the values of sine, cosine, and tangent are as follows:
- Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
- Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
- The sine and cosine functions are periodic with a period of 360°.
- To find the equivalent angle for a given trigonometric ratio, we can use the periodic properties of the sine and cosine functions.
Frequently Asked Questions
Q: What is a trigonometric circle?
A: A trigonometric circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
Q: What is the first quadrant of the trigonometric circle?
A: The first quadrant of the trigonometric circle is the upper right quadrant, where both x and y coordinates are positive.
Q: What are the trigonometric ratios in the first quadrant?
A: The trigonometric ratios in the first quadrant are positive, and the values of sine, cosine, and tangent are as follows: * Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. * Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. * Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
Q: How do I find the equivalent angle for a given trigonometric ratio?
A: To find the equivalent angle for a given trigonometric ratio, we can use the periodic properties of the sine and cosine functions.
Q: What is the period of the sine and cosine functions?
Introduction
In our previous article, we explored the trigonometric circle and filled in the missing angles for the given trigonometric ratios in the first quadrant. In this article, we will answer some frequently asked questions about the trigonometric circle and ratios.
Q&A
Q: What is the difference between the sine and cosine functions?
A: The sine and cosine functions are both trigonometric functions that describe the relationships between the angles and the sides of a right triangle. The main difference between the two functions is that the sine function describes the ratio of the length of the opposite side to the length of the hypotenuse, while the cosine function describes the ratio of the length of the adjacent side to the length of the hypotenuse.
Q: What is the tangent function?
A: The tangent function is a trigonometric function that describes the ratio of the length of the opposite side to the length of the adjacent side. It is defined as the ratio of the sine and cosine functions, i.e., tan(x) = sin(x) / cos(x).
Q: How do I find the value of a trigonometric ratio?
A: To find the value of a trigonometric ratio, you can use a calculator or a trigonometric table. Alternatively, you can use the unit circle to find the values of the trigonometric ratios.
Q: What is the unit circle?
A: The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is used to visualize the relationships between the angles and the trigonometric ratios.
Q: How do I use the unit circle to find the values of the trigonometric ratios?
A: To use the unit circle to find the values of the trigonometric ratios, you need to locate the point on the unit circle that corresponds to the given angle. The coordinates of this point will give you the values of the sine, cosine, and tangent functions.
Q: What is the range of the sine and cosine functions?
A: The range of the sine and cosine functions is [-1, 1]. This means that the values of the sine and cosine functions will always be between -1 and 1.
Q: What is the range of the tangent function?
A: The range of the tangent function is all real numbers. This means that the values of the tangent function can be any real number.
Q: How do I find the inverse of a trigonometric function?
A: To find the inverse of a trigonometric function, you need to swap the x and y coordinates of the function. For example, the inverse of the sine function is the arcsine function, which is defined as sin^(-1)(x) = arcsin(x).
Q: What is the difference between the sine and cosine functions in the second quadrant?
A: In the second quadrant, the sine function is positive, while the cosine function is negative. This is because the x-coordinate is negative, while the y-coordinate is positive.
Q: What is the difference between the sine and cosine functions in the third quadrant?
A: In the third quadrant, both the sine and cosine functions are negative. This is because both the x and y coordinates are negative.
Q: What is the difference between the sine and cosine functions in the fourth quadrant?
A: In the fourth quadrant, the sine function is negative, while the cosine function is positive. This is because the x-coordinate is positive, while the y-coordinate is negative.
Conclusion
In this article, we answered some frequently asked questions about the trigonometric circle and ratios. We hope that this article has been helpful in clarifying any doubts you may have had about these topics.
Key Takeaways
- The sine and cosine functions are both trigonometric functions that describe the relationships between the angles and the sides of a right triangle.
- The tangent function is a trigonometric function that describes the ratio of the length of the opposite side to the length of the adjacent side.
- The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
- The range of the sine and cosine functions is [-1, 1].
- The range of the tangent function is all real numbers.
- The inverse of a trigonometric function is found by swapping the x and y coordinates of the function.
Q: What is the difference between the sine and cosine functions in the second quadrant?
A: In the second quadrant, the sine function is positive, while the cosine function is negative.
Q: What is the difference between the sine and cosine functions in the third quadrant?
A: In the third quadrant, both the sine and cosine functions are negative.
Q: What is the difference between the sine and cosine functions in the fourth quadrant?
A: In the fourth quadrant, the sine function is negative, while the cosine function is positive.
Q: How do I find the value of a trigonometric ratio?
A: To find the value of a trigonometric ratio, you can use a calculator or a trigonometric table. Alternatively, you can use the unit circle to find the values of the trigonometric ratios.
Q: What is the unit circle?
A: The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane.
Q: How do I use the unit circle to find the values of the trigonometric ratios?
A: To use the unit circle to find the values of the trigonometric ratios, you need to locate the point on the unit circle that corresponds to the given angle. The coordinates of this point will give you the values of the sine, cosine, and tangent functions.
Q: What is the range of the sine and cosine functions?
A: The range of the sine and cosine functions is [-1, 1].
Q: What is the range of the tangent function?
A: The range of the tangent function is all real numbers.
Q: How do I find the inverse of a trigonometric function?
A: To find the inverse of a trigonometric function, you need to swap the x and y coordinates of the function. For example, the inverse of the sine function is the arcsine function, which is defined as sin^(-1)(x) = arcsin(x).