What are the steps to getting the tan of 345degree, without the use of a calculator?
345°=360°-15°
tan(A-B)=(tan A - tan B)/(1 - (tan A)*(tan B))
tan(360°)=0
tan(360°-15°)= tan(360°)-tan(15°)/(1-(tan(360°)*(tan(15°))
tan(360°-15°)= 0-tan(15°)/(1-0*(tan(15°))
tan(360°-15°)=tan(345°)= -tan(15°)/(1-0)
tan(345°)= -tan(15°)/1
tan(345°)= -tan(15°)
tan(x/2) = [1 - cos(x)]/sin(x)
tan(30°/2)=tan(15°)
sin(30°)= 1/2
cos(30°)= sqroot(3)/2
tan(30°/2)=tan(15°)= [1 - cos(30°)]/sin(30°)
= (1 - sqroot(3)/2)/(1/2)
= 2 - sqroot(3)
tan(15°) = 2 - sqroot(3)
tan(345°)= -tan(15°)
tan(345°)= -(2 - sqroot(3))
tan(345°)= sqroot(3)-2
or ...
345° is coterminal with -15°
so tan 345° = - tan15
= - tan(45-30)
= - (tan45 - tan30)/(1 + tan45tan30)
= -(1-1/√3)/(1+1/√3) , multiply top and bottom by √3
= (1 - √3)(√3 + 1)
You can rationalize if necessary to get Bosnian's answer
To find the tangent of 345 degrees without using a calculator, you can use the following steps:
1. Understand the reference angle: The reference angle is the acute angle formed between the initial side and the terminal side in standard position. To determine the reference angle for 345 degrees, subtract multiples of 360 degrees from it until you obtain an angle between 0 and 360 degrees.
345 - 360 = -15 (negative angle)
-15 + 360 = 345 (positive angle)
The reference angle for 345 degrees is 15 degrees.
2. Determine the quadrant: By analyzing which quadrant the angle lies in, you can determine the sign of the trigonometric function. In this case, 345 degrees falls in the fourth quadrant, where the tangent function is positive.
3. Recall trigonometric ratios: The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side in a right triangle.
4. Create a right triangle: Since we're working with an angle in standard position, we can create a right triangle with the reference angle of 15 degrees. The opposite side will be 'y' (unknown), the adjacent side will be 'x' (unknown), and the hypotenuse will be 'r' (unknown).
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15°
5. Apply the tangent function: The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, tan(15°) = y/x.
6. Solve for y: Rearranging the equation, we have y = x * tan(15°).
7. Use trigonometric tables or reference angles: Trigonometric tables or reference angles can provide the approximate values for the tangent of specific angles. For example, the tangent of 15 degrees is approximately 0.268.
Therefore, substituting the value of tan(15°) into the equation, we have y = 0.268 * x.
Please note that the above steps provide an approximation and the actual tangent of 345 degrees might be slightly different due to rounding.