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.
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.
We are given the equation: x tan 45° cos 60° = sin 60° cot 60°. Step 1: Substitute the trigonometric values Using standard trigonometric values: - tan 45° = 1, - cos 60° = 1/2, - sin 60° = √3/2, - cot 60° = 1/√3. Substitute these values into the equation: x (1) (1/2) = (√3/2) (1/√3). Step 2: SimplifRead more
We are given the equation:
x tan 45° cos 60° = sin 60° cot 60°.
Step 1: Substitute the trigonometric values
Using standard trigonometric values:
– tan 45° = 1,
– cos 60° = 1/2,
– sin 60° = √3/2,
– cot 60° = 1/√3.
Substitute these values into the equation:
x (1) (1/2) = (√3/2) (1/√3).
Step 2: Simplify both sides
Simplify the left-hand side:
x (1/2) = x/2.
Step 3: Solve for x
Multiply through by 2 to isolate x:
x = 1.
Step 4: Final Answer
The value of x is 1.
The correct answer is:
a) 1
This question pertains to Chapter 8 of the Class 10th NCERT Mathematics textbook, which introduces the topic of Trigonometry. Provide the answer based on your understanding of the chapter.
We are given: 8 tan x = 15. Step 1: Solve for tan x Rearrange the equation to solve for tan x: tan x = 15/8. Step 2: Express sin x and cos x in terms of tan x Using the identity tan x = sin x / cos x, we can write: sin x = 15k and cos x = 8k, where k is a positive constant such that sin²x + cos²x =Read more
We are given:
8 tan x = 15.
Step 1: Solve for tan x
Rearrange the equation to solve for tan x:
tan x = 15/8.
Step 2: Express sin x and cos x in terms of tan x
Using the identity tan x = sin x / cos x, we can write:
sin x = 15k and cos x = 8k,
where k is a positive constant such that sin²x + cos²x = 1 (Pythagorean identity).
Substitute sin x = 15k and cos x = 8k into the identity:
(15k)² + (8k)² = 1
225k² + 64k² = 1
289k² = 1
k² = 1/289
k = √(1/289)
k = 1/17.
Thus:
sin x = 15k = 15/17,
cos x = 8k = 8/17.
Step 3: Find sin x – cos x
Now, calculate sin x – cos x:
sin x – cos x = (15/17) – (8/17)
= (15 – 8)/17
= 7/17.
Step 4: Final Answer
The value of sin x – cos x is 7/17.
The correct answer is:
d) 7/17
This question related to Chapter 8 Mathematics Class 10th NCERT. From the Chapter 8 Introduction to Trigonometry. Give answer according to your understanding.
Finding the Distance of a Point from the x-Axis We are given a point P(2,3) and need to find its distance from the x-axis. Step 1: Understanding Distance from the x-Axis The x-axis is the horizontal axis in the Cartesian plane. The distance of any point (x, y) from the x-axis is simply the absoluteRead more
Finding the Distance of a Point from the x-Axis
We are given a point P(2,3) and need to find its distance from the x-axis.
Step 1: Understanding Distance from the x-Axis
The x-axis is the horizontal axis in the Cartesian plane.
The distance of any point (x, y) from the x-axis is simply the absolute value of its y-coordinate.
This is because the x-axis is located at y=0, and the vertical distance from any point to this axis is determined by how far its y-value is from zero.
Step 2: Applying the Formula
For a point (x,y), the distance from the x-axis is given by:
Distance = |y|
For point P(2, 3), we substitute y = 3:
Distance = |3| = 3
Step 3: Final Answer
Thus, the distance of point P(2,3) from the x-axis is 3.
Correct Option: (b) 3
This question related to Chapter 7 Mathematics Class 10th NCERT. From the Chapter 7 Coordinate Geometry. Give answer according to your understanding.
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.
For more please visit here:
See lesshttps://www.tiwariacademy.in/ncert-solutions/class-10/maths/
If x tan 45° cos 60° = sin 60° cot 60°, then x is equal to
We are given the equation: x tan 45° cos 60° = sin 60° cot 60°. Step 1: Substitute the trigonometric values Using standard trigonometric values: - tan 45° = 1, - cos 60° = 1/2, - sin 60° = √3/2, - cot 60° = 1/√3. Substitute these values into the equation: x (1) (1/2) = (√3/2) (1/√3). Step 2: SimplifRead more
We are given the equation:
x tan 45° cos 60° = sin 60° cot 60°.
Step 1: Substitute the trigonometric values
Using standard trigonometric values:
– tan 45° = 1,
– cos 60° = 1/2,
– sin 60° = √3/2,
– cot 60° = 1/√3.
Substitute these values into the equation:
x (1) (1/2) = (√3/2) (1/√3).
Step 2: Simplify both sides
Simplify the left-hand side:
x (1/2) = x/2.
Simplify the right-hand side:
(√3/2) (1/√3) = (√3 / √3) / 2 = 1/2.
Thus, the equation becomes:
x/2 = 1/2.
Step 3: Solve for x
Multiply through by 2 to isolate x:
x = 1.
Step 4: Final Answer
The value of x is 1.
The correct answer is:
a) 1
This question pertains to Chapter 8 of the Class 10th NCERT Mathematics textbook, which introduces the topic of Trigonometry. Provide the answer based on your understanding of the chapter.
For more please visit here:
See lesshttps://www.tiwariacademy.in/ncert-solutions-class-10-maths-chapter-8/
If 8 tan x = 15, then sin x – cos x is equal to
We are given: 8 tan x = 15. Step 1: Solve for tan x Rearrange the equation to solve for tan x: tan x = 15/8. Step 2: Express sin x and cos x in terms of tan x Using the identity tan x = sin x / cos x, we can write: sin x = 15k and cos x = 8k, where k is a positive constant such that sin²x + cos²x =Read more
We are given:
8 tan x = 15.
Step 1: Solve for tan x
Rearrange the equation to solve for tan x:
tan x = 15/8.
Step 2: Express sin x and cos x in terms of tan x
Using the identity tan x = sin x / cos x, we can write:
sin x = 15k and cos x = 8k,
where k is a positive constant such that sin²x + cos²x = 1 (Pythagorean identity).
Substitute sin x = 15k and cos x = 8k into the identity:
(15k)² + (8k)² = 1
225k² + 64k² = 1
289k² = 1
k² = 1/289
k = √(1/289)
k = 1/17.
Thus:
sin x = 15k = 15/17,
cos x = 8k = 8/17.
Step 3: Find sin x – cos x
Now, calculate sin x – cos x:
sin x – cos x = (15/17) – (8/17)
= (15 – 8)/17
= 7/17.
Step 4: Final Answer
The value of sin x – cos x is 7/17.
The correct answer is:
d) 7/17
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-chapter-8/
The distance of the point P(2, 3) from the x- axis is
Finding the Distance of a Point from the x-Axis We are given a point P(2,3) and need to find its distance from the x-axis. Step 1: Understanding Distance from the x-Axis The x-axis is the horizontal axis in the Cartesian plane. The distance of any point (x, y) from the x-axis is simply the absoluteRead more
Finding the Distance of a Point from the x-Axis
We are given a point P(2,3) and need to find its distance from the x-axis.
Step 1: Understanding Distance from the x-Axis
The x-axis is the horizontal axis in the Cartesian plane.
The distance of any point (x, y) from the x-axis is simply the absolute value of its y-coordinate.
This is because the x-axis is located at y=0, and the vertical distance from any point to this axis is determined by how far its y-value is from zero.
Step 2: Applying the Formula
For a point (x,y), the distance from the x-axis is given by:
Distance = |y|
For point P(2, 3), we substitute y = 3:
Distance = |3| = 3
Step 3: Final Answer
Thus, the distance of point P(2,3) from the x-axis is 3.
Correct Option: (b) 3
This question related to Chapter 7 Mathematics Class 10th NCERT. From the Chapter 7 Coordinate Geometry. Give answer according to your understanding.
For more please visit here:
See lesshttps://www.tiwariacademy.in/ncert-solutions-class-10-maths-chapter-7/