use exact values of the sin,cos and tan of (pi/3) and (pi/6)(which I have found)and the symmetry of the graphs of sin,cos and tan to find the exact values of sin(-pi/6), cos(5pi/3) and tan (4pi/3). I know that sin -(pi/6) = -1/2 and cos (5pi/30 = 1/2 and tan 4pi/3 = root 3. Please will someone tell me what the symmetry of the graphs is and how do I plot them in excel
The symmetry of the graphs of sine, cosine, and tangent can be represented by the following properties:
1. Symmetry of Sine Function: The sine function is an odd function, which means it exhibits symmetry about the origin (0,0). This can be shown by the equation sin(-θ) = -sin(θ), where θ represents an angle.
2. Symmetry of Cosine Function: The cosine function is an even function, which means it exhibits symmetry about the y-axis. This can be shown by the equation cos(-θ) = cos(θ), where θ represents an angle.
3. Symmetry of Tangent Function: The tangent function is an odd function, which means it exhibits symmetry about the origin (0,0). This can be shown by the equation tan(-θ) = -tan(θ), where θ represents an angle.
To plot the graphs of sine, cosine, and tangent in Excel, you can follow these steps:
1. Open a new Excel worksheet.
2. Create a column for the x-values (angles). Enter the desired range of angles for which you want to plot the functions.
3. In the adjacent columns, calculate the corresponding y-values using the following formulas:
- For the sine function, use the formula "=SIN(angle)".
- For the cosine function, use the formula "=COS(angle)".
- For the tangent function, use the formula "=TAN(angle)".
4. Fill down the formulas for the entire range of angles.
5. Create a scatter plot by selecting the x-values and y-values columns, then go to the "Insert" tab and choose the desired scatter plot type.
6. Format the chart as needed, including labeling the axes and adding a title.
To find the exact values of sin(-π/6), cos(5π/3), and tan(4π/3) using the symmetry of the functions:
1. For sin(-π/6): Since sine is an odd function, it exhibits symmetry about the origin. We already know that sin(π/6) = 1/2. Therefore, sin(-π/6) will have the same magnitude but opposite sign, so sin(-π/6) = -1/2.
2. For cos(5π/3): Since cosine is an even function, it exhibits symmetry about the y-axis. We already know that cos(π/3) = 1/2. Therefore, cos(5π/3) will have the same magnitude but the same sign, so cos(5π/3) = 1/2.
3. For tan(4π/3): Since tangent is an odd function, it exhibits symmetry about the origin. We already know that tan(π/3) = √3. Therefore, tan(4π/3) will have the same magnitude but the opposite sign, so tan(4π/3) = -√3.
Hence, sin(-π/6) = -1/2, cos(5π/3) = 1/2, and tan(4π/3) = -√3.