sin 60⁰
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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![### Understanding Sine of 60 Degrees
In trigonometry, the sine of an angle is a function that represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.
Here, we are given the expression \( \sin 60^\circ \).
#### Value of \( \sin 60^\circ \)
The value of \( \sin 60^\circ \) is well-known in trigonometry. It is derived from an equilateral triangle split into two 30-60-90 triangles.
- The exact value of \( \sin 60^\circ \) is \( \frac{\sqrt{3}}{2} \).
This value is crucial in various applications of trigonometry, including solving triangles, physics problems, and engineering calculations.
#### Visual Representation
If we draw a 30-60-90 triangle:
- The side ratios in this triangle are \( 1 : \sqrt{3} : 2 \).
- The side opposite the 60-degree angle (relative to the unit circle or a right triangle) is \( \sqrt{3} \).
- The hypotenuse is 2.
So the sine function, which is the ratio of the opposite side to the hypotenuse, gives us:
\[ \sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \]
Understanding these basic trigonometric functions and their values at specific angles is fundamental in mathematics and its various applications.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee38ad10-d3a7-46a8-a0c4-2e1f024f265a%2Fda0bb8d2-2537-43fa-b6b4-96fd0548fe2c%2F641g9z_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Sine of 60 Degrees
In trigonometry, the sine of an angle is a function that represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right-angled triangle.
Here, we are given the expression \( \sin 60^\circ \).
#### Value of \( \sin 60^\circ \)
The value of \( \sin 60^\circ \) is well-known in trigonometry. It is derived from an equilateral triangle split into two 30-60-90 triangles.
- The exact value of \( \sin 60^\circ \) is \( \frac{\sqrt{3}}{2} \).
This value is crucial in various applications of trigonometry, including solving triangles, physics problems, and engineering calculations.
#### Visual Representation
If we draw a 30-60-90 triangle:
- The side ratios in this triangle are \( 1 : \sqrt{3} : 2 \).
- The side opposite the 60-degree angle (relative to the unit circle or a right triangle) is \( \sqrt{3} \).
- The hypotenuse is 2.
So the sine function, which is the ratio of the opposite side to the hypotenuse, gives us:
\[ \sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} \]
Understanding these basic trigonometric functions and their values at specific angles is fundamental in mathematics and its various applications.
Transcribed Image Text:The text in the image reads as follows:
"cos -30°"
This term represents the cosine of a negative angle, specifically -30 degrees. In trigonometry, the cosine function is an even function, meaning that cos(θ) = cos(-θ). Thus, the value of cos(-30°) is the same as the value of cos(30°).
To calculate the cosine of -30 degrees, one can use the unit circle or trigonometric identities. Here, you will find that:
cos(-30°) = cos(30°) = √3/2
This concept is crucial in understanding the properties of trigonometric functions and their symmetrical behavior around the y-axis.
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