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If a cos θ + b sin θ = m and a sin θ – b cos θ = n, then a² + b² =
We are given the equations: a cosθ + b sinθ = m — (1) a sinθ - b cosθ = n — (2) We need to find the value of a² + b². Step 1: Square both equations Square both sides of equation (1): (a cosθ + b sinθ)² = m² Expand the left-hand side: a²cos²θ + 2ab cosθ sinθ + b²sin²θ = m² — (3) Square both sides ofRead more
We are given the equations:
a cosθ + b sinθ = m — (1)
a sinθ – b cosθ = n — (2)
We need to find the value of a² + b².
Step 1: Square both equations
Square both sides of equation (1):
(a cosθ + b sinθ)² = m²
Expand the left-hand side:
a²cos²θ + 2ab cosθ sinθ + b²sin²θ = m² — (3)
Square both sides of equation (2):
(a sinθ – b cosθ)² = n²
Expand the left-hand side:
a²sin²θ – 2ab sinθ cosθ + b²cos²θ = n² — (4)
Step 2: Add equations (3) and (4)
Add the expanded forms of equations (3) and (4):
(a²cos²θ + 2ab cosθ sinθ + b²sin²θ) + (a²sin²θ – 2ab sinθ cosθ + b²cos²θ) = m² + n²
Simplify the terms:
– The terms involving 2ab cosθ sinθ cancel out.
– Combine the remaining terms:
a²(cos²θ + sin²θ) + b²(sin²θ + cos²θ) = m² + n²
Step 3: Use the Pythagorean identity
From the Pythagorean identity, we know:
cos²θ + sin²θ = 1.
Substitute this into the equation:
a²(1) + b²(1) = m² + n²
Simplify:
a² + b² = m² + n².
The question is based on Chapter 8 of the Class 10th NCERT Mathematics textbook, titled “Introduction to Trigonometry.” Provide your response in line with the concepts covered in this chapter.
For more please visit here:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
If sin θ + sin² θ = 1, then cos² θ + cos⁴ θ =
We are given the equation: sin θ + sin² θ = 1. We need to find the value of cos² θ + cos⁴ θ. Step 1: Express sin² θ in terms of sin θ Rearrange the given equation: sin² θ = 1 - sin θ. Step 2: Use the Pythagorean identity From the Pythagorean identity, we know: sin² θ + cos² θ = 1. Substitute sin² θRead more
We are given the equation:
sin θ + sin² θ = 1.
We need to find the value of cos² θ + cos⁴ θ.
Step 1: Express sin² θ in terms of sin θ
Rearrange the given equation:
sin² θ = 1 – sin θ.
Step 2: Use the Pythagorean identity
From the Pythagorean identity, we know:
sin² θ + cos² θ = 1.
Substitute sin² θ = 1 – sin θ into the identity:
(1 – sin θ) + cos² θ = 1.
Simplify:
cos² θ = sin θ.
Step 3: Express cos⁴ θ in terms of cos² θ
Since cos² θ = sin θ, we can write:
cos⁴ θ = (cos² θ)² = (sin θ)² = sin² θ.
Step 4: Substitute into cos² θ + cos⁴ θ
Now substitute cos² θ = sin θ and cos⁴ θ = sin² θ into the expression cos² θ + cos⁴ θ:
cos² θ + cos⁴ θ = sin θ + sin² θ.
From the given equation, we know:
sin θ + sin² θ = 1.
Thus:
cos² θ + cos⁴ θ = 1.
This question related to Chapter 8 Mathematics Class 10th NCERT. From the Chapter 8 Introduction to Trigonometry. Give answer according to your understanding.
For more please visit here:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q, then p² – q² =
We are given the equations: a cotθ + b cosecθ = p — (1) b cotθ + a cosecθ = q — (2) We need to find the value of p² - q². Step 1: Recall the identity for p² - q² The difference of squares formula states: p² - q² = (p + q)(p - q). Thus, we first calculate p + q and p - q. Step 2: Add equations (1) anRead more
We are given the equations:
a cotθ + b cosecθ = p — (1)
b cotθ + a cosecθ = q — (2)
We need to find the value of p² – q².
Step 1: Recall the identity for p² – q²
The difference of squares formula states:
p² – q² = (p + q)(p – q).
Thus, we first calculate p + q and p – q.
Step 2: Add equations (1) and (2)
Add the two equations:
(a cotθ + b cosecθ) + (b cotθ + a cosecθ) = p + q.
Simplify:
(a + b)cotθ + (a + b)cosecθ = p + q.
Factor out (a + b):
(a + b)(cotθ + cosecθ) = p + q. — (3)
Step 3: Subtract equation (2) from (1)
Subtract equation (2) from equation (1):
(a cotθ + b cosecθ) – (b cotθ + a cosecθ) = p – q.
Simplify:
(a – b)cotθ + (b – a)cosecθ = p – q.
Factor out (a – b):
(a – b)(cotθ – cosecθ) = p – q. — (4)
Step 4: Substitute into p² – q²
Using the identity p² – q² = (p + q)(p – q), substitute the expressions for p + q and p – q from equations (3) and (4):
p² – q² = [(a + b)(cotθ + cosecθ)] × [(a – b)(cotθ – cosecθ)].
Simplify:
p² – q² = (a + b)(a – b) × (cotθ + cosecθ)(cotθ – cosecθ).
Step 5: Simplify further using the difference of squares
The term (cotθ + cosecθ)(cotθ – cosecθ) is a difference of squares:
(cotθ + cosecθ)(cotθ – cosecθ) = cot²θ – cosec²θ.
From the trigonometric identity:
cosec²θ – cot²θ = 1,
we can write:
cot²θ – cosec²θ = -1.
Substitute this into the expression for p² – q²:
p² – q² = (a + b)(a – b) × (-1).
Simplify:
p² – q² = -(a + b)(a – b).
Expand (a + b)(a – b) using the difference of squares:
(a + b)(a – b) = a² – b².
Thus:
p² – q² = -(a² – b²).
This question is related to Chapter 8, “Introduction to Trigonometry,” from the Class 10th NCERT Mathematics textbook. Answer the question using your knowledge and understanding of the chapter.
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variable capacitor is connected to a 200 V battery. If its capacitance is changed from 2 µF to X µF, the decrease in energy of the capacitor is 2 X 10⁻² J. The value of X is
The correct answer is (a) 1 µF. The energy stored in a capacitor is given by U = 1/2CV². The decrease in energy is: ΔU = 1/2(2 × 10⁻⁶ - X) (200)² = 2 × 10⁻²J Solving for X, we get 1 µF. For more visit here: https://www.tiwariacademy.com/ncert-solutions/class-12/physics/chapter-2/
The correct answer is (a) 1 µF.
The energy stored in a capacitor is given by
U = 1/2CV². The decrease in energy is:
ΔU = 1/2(2 × 10⁻⁶ – X) (200)² = 2 × 10⁻²J
Solving for X, we get 1 µF.
For more visit here:
See lesshttps://www.tiwariacademy.com/ncert-solutions/class-12/physics/chapter-2/
Why is it important to handle kitchen tools safely?
Handling kitchen tools safely is essential to prevent injuries and ensure hygiene. Knives should be used with caution, keeping fingers away from the blade. Graters, peelers and mixers should be handled carefully to avoid cuts. Using clean surfaces and properly storing tools prevent contamination. FoRead more
Handling kitchen tools safely is essential to prevent injuries and ensure hygiene. Knives should be used with caution, keeping fingers away from the blade. Graters, peelers and mixers should be handled carefully to avoid cuts. Using clean surfaces and properly storing tools prevent contamination. Following safety guidelines ensures smooth food preparation, reduces risks and maintains the quality of ingredients. Safe handling practices also promote confidence and efficiency in the kitchen.
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