Find the actual value of
Cos45+sin60-tan30 by showing all our working out
These are standard angles, whose functions you would do well to learn
Cos45+sin60-tan30
= 1/√2 + √3/2 - 1/√3
= (3√2 + √3)/6
Ttui
To find the actual value of the expression cos(45) + sin(60) - tan(30), we will use the trigonometric values of the angles involved. Here's how you can calculate each term and then combine them:
1. Cosine of 45 degrees (cos(45)):
The cosine of 45 degrees is √2/2. This can be determined from the unit circle or by using trigonometric identities.
2. Sine of 60 degrees (sin(60)):
The sine of 60 degrees is √3/2. This is a commonly known value that can be found in reference tables or by using the unit circle.
3. Tangent of 30 degrees (tan(30)):
To find the tangent of 30 degrees, we take the ratio of the sine and cosine of 30 degrees. The sine of 30 degrees is 1/2, and the cosine of 30 degrees is √3/2. Hence, tan(30) = (1/2) / (√3/2) = 1 / √3. To simplify further, we multiply both the numerator and denominator by √3 to get (1 * √3) / (√3 * √3) = √3 / 3.
Now, let's substitute the values we found into the given expression:
cos(45) + sin(60) - tan(30) = (√2/2) + (√3/2) - (√3/3)
To combine the terms, we need to find a common denominator:
Since (√2/2) and (√3/2) have the same denominator, we can add them directly.
(√2/2) + (√3/2) - (√3/3) = (√2 + √3) / 2 - (√3/3)
To subtract (√3/3), we need a common denominator of 2.
Multiplying (√3/3) by 2/2, we get (2√3/6) = (√3/3).
Now, we can rewrite the expression with a common denominator:
(√2 + √3)/2 - (√3/3) = ((√2 + √3) - (√3)) / 2 = (√2)/2
Therefore, the actual value of cos(45) + sin(60) - tan(30) is (√2)/2.