Answer:
Exercise 8.1 Page: 181
1. In ∆ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) sin A, cos A
(ii) sin C, cos C
Solution:
In a given triangle ABC, right angled at B = ∠B = 90°
Given: AB = 24 cm and BC = 7 cm
According to the Pythagoras Theorem,
In a right- angled triangle, the squares of the hypotenuse side is equal to the sum of the squares of the other two sides.
By applying Pythagoras theorem, we get
AC2=AB2+BC2
AC2 = (24)2+72
AC2 = (576+49)
AC2 = 625cm2
AC = √625 = 25
Therefore, AC = 25 cm
(i) To find Sin (A), Cos (A)
We know that sine (or) Sin function is the equal to the ratio of length of the opposite side to the hypotenuse side. So it becomes
Sin (A) = Opposite side /Hypotenuse = BC/AC = 7/25
Cosine or Cos function is equal to the ratio of the length of the adjacent side to the hypotenuse side and it becomes,
Cos (A) = Adjacent side/Hypotenuse = AB/AC = 24/25
Step-by-step explanation:
sorry I can help you this only
Let's simplify the given equation step by step:
costheta / (1 - sintheta) + costheta / (1 + sintheta) = 4
To combine the two fractions, we need to find a common denominator, which is (1 - sintheta)(1 + sintheta):
[(costheta)(1 + sintheta) + (costheta)(1 - sintheta)] / [(1 - sintheta)(1 + sintheta)] = 4
Expanding the numerator:
[ costheta + costheta sintheta + costheta - costheta sintheta ] / [(1 - sintheta)(1 + sintheta)] = 4
Combining like terms:
[2costheta] / [(1 - sintheta)(1 + sintheta)] = 4
Now, let's simplify the denominator:
(1 - sintheta)(1 + sintheta) = 1 - sin^2(theta) [Using the identity (a + b)(a - b) = a^2 - b^2]
(1 - sin^2(theta)) = cos^2(theta) [Using the identity sin^2(theta) + cos^2(theta) = 1]
Plugging this back into the equation:
[2costheta] / cos^2(theta) = 4
Multiplying both sides by cos^2(theta):
2costheta = 4cos^2(theta)
Dividing both sides by 2:
costheta = 2cos^2(theta)
Using the identity cos^2(theta) = 1 - sin^2(theta):
costheta = 2(1 - sin^2(theta))
Rearranging the equation:
2sin^2(theta) + costheta - 2 = 0
Now, we have a quadratic equation in terms of sin(theta). To solve this equation, we can use the quadratic formula:
sin(theta) = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 1, and c = -2. Plugging in these values:
sin(theta) = (-(1) ± √((1)^2 - 4(2)(-2))) / (2(2))
sin(theta) = (-1 ± √(1 + 16)) / 4
sin(theta) = (-1 ± √17) / 4
Since theta is less than 90 degrees, sin(theta) cannot be greater than 1. Therefore, we take the positive square root:
sin(theta) = (-1 + √17) / 4
To find the value of theta, we take the inverse sine (sin^(-1)) of both sides:
theta = sin^(-1)((-1 + √17) / 4)
Using a calculator, the approximate value of theta is 37.09 degrees.
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Answers & Comments
Answer:
Exercise 8.1 Page: 181
1. In ∆ ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine:
(i) sin A, cos A
(ii) sin C, cos C
Solution:
In a given triangle ABC, right angled at B = ∠B = 90°
Given: AB = 24 cm and BC = 7 cm
According to the Pythagoras Theorem,
In a right- angled triangle, the squares of the hypotenuse side is equal to the sum of the squares of the other two sides.
By applying Pythagoras theorem, we get
AC2=AB2+BC2
AC2 = (24)2+72
AC2 = (576+49)
AC2 = 625cm2
AC = √625 = 25
Therefore, AC = 25 cm
(i) To find Sin (A), Cos (A)
We know that sine (or) Sin function is the equal to the ratio of length of the opposite side to the hypotenuse side. So it becomes
Sin (A) = Opposite side /Hypotenuse = BC/AC = 7/25
Cosine or Cos function is equal to the ratio of the length of the adjacent side to the hypotenuse side and it becomes,
Cos (A) = Adjacent side/Hypotenuse = AB/AC = 24/25
Step-by-step explanation:
sorry I can help you this only
Answer:
Let's simplify the given equation step by step:
costheta / (1 - sintheta) + costheta / (1 + sintheta) = 4
To combine the two fractions, we need to find a common denominator, which is (1 - sintheta)(1 + sintheta):
[(costheta)(1 + sintheta) + (costheta)(1 - sintheta)] / [(1 - sintheta)(1 + sintheta)] = 4
Expanding the numerator:
[ costheta + costheta sintheta + costheta - costheta sintheta ] / [(1 - sintheta)(1 + sintheta)] = 4
Combining like terms:
[2costheta] / [(1 - sintheta)(1 + sintheta)] = 4
Now, let's simplify the denominator:
(1 - sintheta)(1 + sintheta) = 1 - sin^2(theta) [Using the identity (a + b)(a - b) = a^2 - b^2]
(1 - sin^2(theta)) = cos^2(theta) [Using the identity sin^2(theta) + cos^2(theta) = 1]
Plugging this back into the equation:
[2costheta] / cos^2(theta) = 4
Multiplying both sides by cos^2(theta):
2costheta = 4cos^2(theta)
Dividing both sides by 2:
costheta = 2cos^2(theta)
Using the identity cos^2(theta) = 1 - sin^2(theta):
costheta = 2(1 - sin^2(theta))
Rearranging the equation:
2sin^2(theta) + costheta - 2 = 0
Now, we have a quadratic equation in terms of sin(theta). To solve this equation, we can use the quadratic formula:
sin(theta) = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 2, b = 1, and c = -2. Plugging in these values:
sin(theta) = (-(1) ± √((1)^2 - 4(2)(-2))) / (2(2))
sin(theta) = (-1 ± √(1 + 16)) / 4
sin(theta) = (-1 ± √17) / 4
Since theta is less than 90 degrees, sin(theta) cannot be greater than 1. Therefore, we take the positive square root:
sin(theta) = (-1 + √17) / 4
To find the value of theta, we take the inverse sine (sin^(-1)) of both sides:
theta = sin^(-1)((-1 + √17) / 4)
Using a calculator, the approximate value of theta is 37.09 degrees.