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You use the definitions of secant (sec) and cosecant (csc):
sec x = 1/cos x
csc x = 1/sin x
→ sec² 45° + csc² 60°
= (1/cos 45°)² + 1/(sin 60°)²
= (1/(1/√2))² + (1/(√3/2))²
= (√2)² + (2/√3)²
= 2 + 4/3
= 10/3 = 3⅓
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In a right-triangle with an angle of 45°, the other angle is 45° and the sides are in the ratio of 1 : 1 : √2
(sides opposite angles 45° : 45° : 90°) giving cos 45° = 1/√2
In a right triangle with an angle of 60°, the other angle is 30° and the sides are in the ratio of √3 : 1 : 2 (sides opposite angles 60° : 30° : 90°) giving sin 60° = √3/2
These ratios can be confirmed/calculated by Pythagoras; in the first case it is an isosceles triangle with legs of length 1 unit, and in the second case it is half an equilateral triangle (one angle bisected by a perpendicular from the opposite side) with side length 2 units.
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Yes.
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They are co-functions meaning that 90 - sec x = csc x.
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tan(x)*csc(x) = sec(x)
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cot(x)=1/tan(x)=1/(sin(x)/cos(x))=cos(x)/sin(x) csc(x)=1/sin(x) sec(x)=1/cos(x) Therefore, (csc(x))2/cot(x)=(1/(sin(x))2)/cot(x)=(1/(sin(x))2)/(cos(x)/sin(x))=(1/(sin(x))2)(sin(x)/cos(x))=(1/sin(x))*(1/cos(x))=csc(x)*sec(x)
What is sin cos tan csc sec cot of 135 degrees?
All those can be calculated quickly with your calculator. Just be sure it is in "degrees" mode (not in radians). Also, use the following identities: csc(x) = 1 / sin(x) sec(x) = 1 / cos(x) cot(x) = 1 / tan(x) or the equivalent cos(x) / sin(x)
Sec squared theta plus tan squared theta equals to 13 12 and sec raised to 4 theta minus tan raised to 4 theta equals to how much?
It also equals 13 12.
What is the second derivative of ln(tan(x))?
f'(x) = 1/tan(x) * sec^2(x) where * means multiply and ^ means to the power of. = cot(x) * sec^2(x) f''(x) = f'(cot(x)*sec^2(x) + cot(x)*f'[sec^2(x)] = -csc^2(x)*sec^2(x) + cot(x)*2tan(x)sec^2(x) = sec^2(x) [cot(x)-csc^2(x)] +2tan(x)cot(x) = sec^2(x) [cot(x)-csc^2(x)] +2
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sin(45) = cos(45) = 1/sqrt(2) tan(45) = cot(45)= 1 csc(45) = sec(45) = sqrt(2)
What is sin cos tan csc sec cot of 30 45 60 degrees?
Sin(30) = 1/2 Sin(45) = root(2)/2 Sin(60) = root(3)/2 Cos(30) = root(3)/2 Cos(45) = root(2)/2 Cos(60) = 1/2 Tan(30) = root(3)/3 Tan(45) = 1 Tan(60) = root(3) Csc(30) = 2 Csc(45) = root(2) Csc(60) = 2root(3)/3 Sec(30) = 2root(3)/3 Sec(45) = root(2) Sec(60) = 2 Cot(30) = root(3) Cot(45) = 1 Cot(60) = root(3)/3
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By converting cosecants and secants to the equivalent sine and cosine functions. For example, csc theta is the same as 1 / sin thetha.
Csc x tan x?
If you want to simplify that, it usually helps to express all the trigonometric functions in terms of sines and cosines.
