Find the exact value of sec 60°tan135°-cot60°sin60°
To find the exact value of the given expression, let's simplify each term step by step:
1. Start with the expression: sec 60°tan135° - cot60°sin60°
2. Let's deal with sec 60° first. Recall that sec is the reciprocal of cosine, so secθ = 1/cosθ. Therefore, sec 60° = 1/cos 60°.
3. Find the cosine of 60°: cos 60° = 1/2.
4. Substitute the value back into the expression: sec 60° = 1/(1/2) = 2.
5. Now, let's tackle the term tan 135°. Recall that tan is the ratio of sine to cosine, so tanθ = sinθ/cosθ. Therefore, tan 135° = sin 135°/cos 135°.
6. Find the sine and cosine of 135°: sin 135° = -sqrt(2)/2 and cos 135° = -sqrt(2)/2.
7. Substitute the values back into the expression: tan 135° = (-sqrt(2)/2)/(-sqrt(2)/2) = 1.
8. Moving on to cot 60°. Recall that cot is the reciprocal of tan, so cotθ = 1/tanθ. Therefore, cot 60° = 1/tan 60°.
9. Find the tangent of 60°: tan 60° = sqrt(3).
10. Substitute the value back into the expression: cot 60° = 1/(sqrt(3)).
11. Lastly, sin 60° = sqrt(3)/2.
12. Substitute all the respective values back into the original expression:
sec 60°tan 135° - cot 60°sin 60°
= (2)(1) - (1/√3)(√3/2)
= 2 - (1/2)
= 4/2 - 1/2
= 3/2.
Therefore, the exact value of sec 60°tan 135° - cot 60°sin 60° is 3/2.
That can be calculated without a 'calculator' or trig tables.
sec 60 = 1/cos 60 = 1/(1/2) = 2
tan 135 = -tan 45 = -1
cot 60 = tan 30 = 1/sqrt3
sin 60 = (sqrt3)/2
Do the calculation.