Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It has numerous applications in various fields, including physics, engineering, and architecture. One concept that is essential to understand in trigonometry is the value of trigonometric functions such as sine, cosine, and tangent. In this article, we will focus specifically on the value of sin 135 and explore its significance.
Sin, short for sine, is a trigonometric function that relates the ratio of the length of the side opposite an angle to the length of the triangle’s hypotenuse. It plays a crucial role in determining the behavior and properties of triangles. Sin 135 refers to the sine value of an angle of 135 degrees.
What is the value of sin 135?
The value of sin 135 is **-√2/2**.
The angle 135 degrees falls in the second quadrant of the coordinate plane, where the sine function is negative. Sin 135 is mathematically calculated by considering a reference angle of 45 degrees (180 degrees – 135 degrees), which corresponds to the positive x-axis in the first quadrant.
By using the definition of the sine function, sin x = opposite/hypotenuse, we can construct a right-angled triangle within the second quadrant to find the value of sin 135. The length of the side opposite to the angle 45 degrees (opposite side) would be √2 and the hypotenuse (the distance from the origin to our point) will be 2. This gives us sin 135 = -√2/2.
Related FAQs:
1. What is trigonometry?
Trigonometry is a branch of mathematics that focuses on the study of angles, triangles, and their relationships.
2. What is a sine function?
The sine function is a trigonometric function that relates the ratio of the length of the side opposite an angle to the length of the triangle’s hypotenuse.
3. What are the key angles in trigonometry?
The key angles in trigonometry are 0°, 30°, 45°, 60°, and 90°, which are commonly referred to as the special angles.
4. How are trigonometric functions represented?
Trigonometric functions are commonly represented using abbreviations such as sin for sine, cos for cosine, and tan for tangent.
5. How is the value of sin calculated for any angle?
The value of sin for any angle can be calculated by forming a right-angled triangle and finding the ratio of the length of the side opposite to the angle to the length of the triangle’s hypotenuse.
6. What are the four quadrants in the coordinate plane?
The four quadrants in the coordinate plane are named as first, second, third, and fourth quadrant, starting from the top right and going counter-clockwise.
7. What is the reference angle?
The reference angle is the positive acute angle formed between the terminal arm (ray extending from the origin to the point on the unit circle) and the x-axis.
8. How can angles greater than 90 degrees be related to angles within a right triangle?
Angles greater than 90 degrees can be related to angles within a right triangle by determining the corresponding reference angle within the first quadrant.
9. What is the range of values for sin?
The range of values for sin is between -1 and 1, inclusive.
10. What is the value of sin 30?
The value of sin 30 is 1/2, which corresponds to the side opposite the angle of 30 degrees divided by the hypotenuse of the right triangle.
11. What is the value of sin 90?
The value of sin 90 is 1, as the side opposite the angle of 90 degrees is equal to the hypotenuse of the right triangle.
12. How is sin 135 related to other trigonometric functions?
Sin 135 can be related to other trigonometric functions by using the identities and relationships derived from the unit circle, such as cos 135 = -√2/2 and tan 135 = 1.