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If cos A + cos²A = 1, then sin² A + sin⁴ A =
We are given the equation: cos A + cos²A = 1. We need to find the value of sin²A + sin⁴A. Step 1: Express cos²A in terms of cos A Rearrange the given equation: cos²A = 1 - cos A. Step 2: Use the Pythagorean identity From the Pythagorean identity, we know: sin²A + cos²A = 1. Substitute cos²A = 1 - coRead more
We are given the equation:
cos A + cos²A = 1.
We need to find the value of sin²A + sin⁴A.
Step 1: Express cos²A in terms of cos A
Rearrange the given equation:
cos²A = 1 – cos A.
Step 2: Use the Pythagorean identity
From the Pythagorean identity, we know:
sin²A + cos²A = 1.
Substitute cos²A = 1 – cos A into the identity:
sin²A + (1 – cos A) = 1.
Simplify:
sin²A = cos A.
Step 3: Express sin⁴A in terms of sin²A
Since sin²A = cos A, we can write:
sin⁴A = (sin²A)² = (cos A)² = cos²A.
Step 4: Substitute into sin²A + sin⁴A
Now substitute sin²A = cos A and sin⁴A = cos²A into the expression sin²A + sin⁴A:
sin²A + sin⁴A = cos A + cos²A.
From the given equation, we know:
cos A + cos²A = 1.
Thus:
sin²A + sin⁴A = 1.
This question is from Chapter 8 of the Class 10th NCERT Mathematics book, which deals with the Introduction to Trigonometry. Answer the question according to your understanding of the chapter.
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Three capacitors 2 pF, 3 pF and 6 pF are joined in series with eachother. The equivalent capacitance is
The correct answer is (d) 1/2 µF. For capacitors in series, the equivalent capacitance is given by: 1/Ceq = 1/C₁ + 1/C₂ + 1/C₃ Substituting C1 = 2 pF, C₂ = 3 pF and C₃ = 6 pF, we get Ceq = 1 pF = 1/2 µF. For more visit here: https://www.tiwariacademy.com/ncert-solutions/class-12/physics/chapter-2/
The correct answer is (d) 1/2 µF.
For capacitors in series, the equivalent capacitance is given by:
1/Ceq = 1/C₁ + 1/C₂ + 1/C₃
Substituting C1 = 2 pF, C₂ = 3 pF and C₃ = 6 pF, we get Ceq = 1 pF = 1/2 µF.
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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.
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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.
For more please visit here:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/