cos 17pi/6 + tan 17pi/6
Perhaps you might find it easier to think in degrees ?
17π/6 radians = 510°
which is coterminal with 150°
cos 150° = - cos30° = -√3/2 , by CAST rule
tan 150 = - tan30 = -1/√3
so cos 17π/6 + tan 17π/6
= -√3/2 - 1√3 , but 1/√3 = √3/3
= -√3/2 - √3/3
= ( -3√3 - 2√3)/6 = -5√3/6
To solve the expression cos(17π/6) + tan(17π/6), we can break it down step by step:
Step 1: Determine the angle
The given angle is 17π/6 which is in radians. We need to convert it to degrees to make it easier to work with.
To convert radians to degrees, we use the formula: radians * (180/π).
17π/6 * (180/π) = 510/6 = 85 degrees
So, the angle is 85 degrees.
Step 2: Calculate cos(85 degrees)
Using a calculator or trigonometric table, we find that cos(85 degrees) ≈ 0.087.
Step 3: Calculate tan(85 degrees)
Using a calculator or trigonometric table, we find that tan(85 degrees) ≈ 11.43.
Step 4: Add the values together
0.087 + 11.43 = 11.517.
So, cos(17π/6) + tan(17π/6) is approximately equal to 11.517.
To find the value of cos(17π/6) + tan(17π/6), we'll first calculate each term separately and then add them together.
1. Finding cos(17π/6):
To calculate the cosine of an angle, we need to determine the reference angle first. For 17π/6, we can identify the reference angle by subtracting it from the nearest multiple of 2π (360 degrees), which is 2π.
Reference angle = 2π - (17π/6)
= (12π/6) - (17π/6)
= -5π/6
Now, we know that the cosine function is positive in the first and fourth quadrants. The angle 17π/6 lies in the third quadrant (180 - 270 degrees), where the cosine function is negative.
cos(17π/6) = -cos(5π/6)
Using the unit circle, we can determine the cosine value for the reference angle of 5π/6, which is √3/2.
Therefore, cos(17π/6) = -√3/2.
2. Finding tan(17π/6):
Tangent is calculated by taking the sine of an angle and dividing it by the cosine of the same angle.
Using the unit circle, we can find the sine and cosine values for the given angle of 17π/6.
sin(17π/6) = sin(-π/6) since sin has a period of 2π
= -sin(π/6)
= -1/2
cos(17π/6) = -√3/2 (from the previous calculation)
Now, we can calculate tan(17π/6) = sin(17π/6) / cos(17π/6) using the above values.
tan(17π/6) = (-1/2) / (-√3/2)
= 1/√3
= √3/3
3. Adding cos(17π/6) and tan(17π/6):
cos(17π/6) + tan(17π/6) = -√3/2 + √3/3
To add these two fractions, we need a common denominator. The least common multiple of 2 and 3 is 6.
Moving the fractions to have a common denominator, we get:
= (-√3 * 3)/(2 * 3) + (√3 * 2)/(3 * 2)
= (-3√3 + 2√3) / 6
= -√3 / 6
Therefore, cos(17π/6) + tan(17π/6) = -√3 / 6.