find the exact value of [sec(-30 degrees)- cot 120 degrees]/1-cosec^2(45 degrees)
Find the exact value of [sin^2(355 degrees) + cos^2 (355 degrees)]/ tan^2 (30 degrees)
sin^2 + cos^2 = 1, so you have
1/tan^2(30) = 1/3
for the other, plug in the values. recall that 1-csc^2 = -cot^2, so the bottom is just -1
To find the exact value of the expression [sec(-30 degrees) - cot(120 degrees)] / [1 - cosec^2(45 degrees)], we'll break it down step by step.
Let's start with the numerator:
1. sec(-30 degrees): Recall that sec(x) is the reciprocal of cosine. Since cosine is an even function, sec(-x) = sec(x). Therefore, sec(-30 degrees) is equal to sec(30 degrees).
2. cot(120 degrees): Recall that cot(x) is equal to the reciprocal of tan(x). Similarly, cot(x + 180 degrees) = cot(x). Therefore, cot(120 degrees) is equal to cot(120 degrees - 180 degrees) = cot(-60 degrees). Since cot(x) is an odd function, cot(-x) = -cot(x). Therefore, cot(-60 degrees) is equal to -cot(60 degrees).
Now, let's simplify the denominator:
3. 1 - cosec^2(45 degrees): Recall that cosec(x) is the reciprocal of sin(x). Since sin(x) is an odd function, cosec(-x) = -cosec(x). Therefore, cosec^2(45 degrees) = cosec^2(-45 degrees), which is equal to -cosec^2(45 degrees).
Now, let's substitute these simplified values back into the expression:
[sec(30 degrees) - (-cot(60 degrees))] / [1 - (-cosec^2(45 degrees))]
[sec(30 degrees) + cot(60 degrees)] / [1 + cosec^2(45 degrees)]
Now, let's find the value of each trigonometric function involved:
4. sec(30 degrees): Since sec(x) is the reciprocal of cosine, sec(30 degrees) is equal to 1 / cos(30 degrees). We can use the special triangle for a 30-60-90 degree triangle to find the cosine of 30 degrees. The triangle has sides in the ratio 1:√3:2, so cos(30 degrees) = √3/2. Therefore, sec(30 degrees) = 1 / (√3/2), which simplifies to 2 / √3. To rationalize the denominator, we multiply by (√3/√3), which gives (2√3) / 3.
5. cot(60 degrees): Since cot(x) is the reciprocal of tan(x), cot(60 degrees) is equal to 1 / tan(60 degrees). The tangent of 60 degrees can be found using the special triangle for a 30-60-90 degree triangle. In this case, the triangle has sides in the ratio 1:√3:2, so tan(60 degrees) = √3/1 = √3. Therefore, cot(60 degrees) = 1 / (√3), which simplifies to √3/3.
6. cosec^2(45 degrees): Since cosec(x) is the reciprocal of sin(x), we can find sin(45 degrees) using the special triangle for a 45-45-90 degree triangle. In this triangle, both legs have a length of 1, so sin(45 degrees) = 1/√2 = √2/2. Therefore, cosec(45 degrees) = 1 / (√2/2), which simplifies to 2 / √2. To rationalize the denominator, we multiply by (√2/√2), which gives (√2)/2. Finally, cosec^2(45 degrees) = (√2/2)^2 = (√2)^2 / (2^2) = 2/4 = 1/2.
Now, substituting these values back into the expression:
[(2√3) / 3 + (√3)/3] / [1 + 1/2]
Simplifying further:
[(3√3) / 3] / [3/2]
Cancelling out the common factor of 3:
[(√3) / 1] / [1/2]
Dividing by a fraction is equivalent to multiplying by its reciprocal:
(√3) / 1 * (2/1)
Multiplying:
2√3
So, the exact value of the expression [sec(-30 degrees) - cot(120 degrees)] / [1 - cosec^2(45 degrees)] is 2√3.
Now, let's move on to the second question:
To find the exact value of [sin^2(355 degrees) + cos^2(355 degrees)] / tan^2(30 degrees), we'll break it down step by step.
1. sin^2(355 degrees): Since the sine function has a period of 360 degrees, we can rewrite 355 degrees as 355 - 360 = -5 degrees. The sine function is an odd function, so sin(-x) = -sin(x). Therefore, sin^2(-5 degrees) = (-sin^2(5 degrees)). Since the sine function is odd, sin^2(5 degrees) = sin^2(-5 degrees). So, sin^2(-5 degrees) = (-sin^2(5 degrees)) = -sin^2(5 degrees).
2. cos^2(355 degrees): Similar to the sine function, the cosine function has a period of 360 degrees. We can rewrite 355 degrees as 355 - 360 = -5 degrees. The cosine function is an even function, so cos(-x) = cos(x). Therefore, cos^2(-5 degrees) = cos^2(5 degrees).
3. tan^2(30 degrees): The tangent of 30 degrees can be found using the special triangle for a 30-60-90 degree triangle. In this case, the triangle has sides in the ratio 1:√3:2, so tan(30 degrees) = √3/1 = √3. Therefore, tan^2(30 degrees) = (√3)^2 = 3.
Now, substituting these values back into the expression:
[-sin^2(5 degrees) + cos^2(5 degrees)] / 3
Since sin^2(x) + cos^2(x) = 1 for any angle x, the expression can be simplified to:
[1 - sin^2(5 degrees)] / 3
And using the trigonometric identity cos^2(x) = 1 - sin^2(x), we can further simplify the expression to:
[cos^2(5 degrees)] / 3
Since cosine is an even function, cos^2(-x) = cos^2(x). Therefore, cos^2(5 degrees) = cos^2(-5 degrees), which can be written as cos^2(355 degrees).
So, the exact value of the expression [sin^2(355 degrees) + cos^2(355 degrees)] / tan^2(30 degrees) is:
[cos^2(355 degrees)] / 3