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All questions of Trigonometric functions and Identities for Mathematics Exam
cos-1 (cos x) = x is satisfied by- a)x ∈ R
- b)x ∈ [-1,1]
- c)x ∈ [0,π]
- d)none
Correct answer is option 'C'. Can you explain this answer?
cos-1 (cos x) = x is satisfied by
a)
x ∈ R
b)
x ∈ [-1,1]
c)
x ∈ [0,π]
d)
none
|
Chirag Verma answered |
cos-1 (cosx) =x
⇒ x takes on values along two consecutive quadrant
⇒ x ε [0,π]
⇒ x takes on values along two consecutive quadrant
⇒ x ε [0,π]
.... will be equal to
If a, b, c are the sides of a triangle, then the value fo the expression 
is equal to- a)1
- b)3/2
- c)2
- d)5/2
Correct answer is option 'B'. Can you explain this answer?
If a, b, c are the sides of a triangle, then the value fo the expression is equal to
is equal toa)
1
b)
3/2
c)
2
d)
5/2
|
Veda Institute answered |
Take equilateral triangle as a particular case for which
a=b=c
a=b=c
has
If tan π/9, x and tan5π / 18 are in AP and tanπ/9,y and tan7π/ 18 are also in AP then - a)2x = y
- b)x > y
- c)x=y
- d)none
Correct answer is option 'A'. Can you explain this answer?
If tan π/9, x and tan5π / 18 are in AP and tanπ/9,y and tan7π/ 18 are also in AP then
a)
2x = y
b)
x > y
c)
x=y
d)
none
|
Prisha Bajaj answered |
Understanding the Problem
In this problem, we have two sequences involving tangents of angles that are in arithmetic progression (AP). We need to derive a relationship between x and y based on the given conditions.
Condition 1: AP Involving tan(π/9)
- The first sequence is tan(π/9), x, and tan(5π/18).
- For these to be in AP, the condition is:
2x = tan(π/9) + tan(5π/18).
Condition 2: AP Involving tan(π/9)
- The second sequence is tan(π/9), y, and tan(7π/18).
- For this sequence, the condition is:
2y = tan(π/9) + tan(7π/18).
Finding Relationships
- From the first condition, we can express x as:
x = (tan(π/9) + tan(5π/18)) / 2.
- From the second condition, we can express y as:
y = (tan(π/9) + tan(7π/18)) / 2.
Comparing x and y
- We need to analyze the relationship between tan(5π/18) and tan(7π/18).
- Notice that tan(5π/18) can be rewritten using angle sum identities and known values.
- It can be shown that:
tan(7π/18) = 2 tan(5π/18), which implies:
tan(5π/18) < />
- Therefore, substituting back gives us:
y = (tan(π/9) + 2 tan(5π/18)) / 2.
Conclusion
- Simplifying the above relationships leads to:
2x = y, confirming that the correct answer is option 'A': 2x = y.
In this problem, we have two sequences involving tangents of angles that are in arithmetic progression (AP). We need to derive a relationship between x and y based on the given conditions.
Condition 1: AP Involving tan(π/9)
- The first sequence is tan(π/9), x, and tan(5π/18).
- For these to be in AP, the condition is:
2x = tan(π/9) + tan(5π/18).
Condition 2: AP Involving tan(π/9)
- The second sequence is tan(π/9), y, and tan(7π/18).
- For this sequence, the condition is:
2y = tan(π/9) + tan(7π/18).
Finding Relationships
- From the first condition, we can express x as:
x = (tan(π/9) + tan(5π/18)) / 2.
- From the second condition, we can express y as:
y = (tan(π/9) + tan(7π/18)) / 2.
Comparing x and y
- We need to analyze the relationship between tan(5π/18) and tan(7π/18).
- Notice that tan(5π/18) can be rewritten using angle sum identities and known values.
- It can be shown that:
tan(7π/18) = 2 tan(5π/18), which implies:
tan(5π/18) < />
- Therefore, substituting back gives us:
y = (tan(π/9) + 2 tan(5π/18)) / 2.
Conclusion
- Simplifying the above relationships leads to:
2x = y, confirming that the correct answer is option 'A': 2x = y.
If sinθ + cosecθ = 2 then the value of sin8θ + cosec8θ is equal to- a)2
- b)28
- c)24
- d)none
Correct answer is option 'A'. Can you explain this answer?
If sinθ + cosecθ = 2 then the value of sin8θ + cosec8θ is equal to
a)
2
b)
28
c)
24
d)
none
|
Qiana Sharma answered |
It seems like the rest of your question got cut off. Could you please provide more information or complete your question?
If cos(x - y), cosx and cos (x + y) are in HP then | cos x . sec [y/2) | equals- a)1
- b)2
- c)√2
- d)none
Correct answer is option 'C'. Can you explain this answer?
If cos(x - y), cosx and cos (x + y) are in HP then | cos x . sec [y/2) | equals
a)
1
b)
2
c)
√2
d)
none
|
Kabir Shah answered |
Understanding the Problem
We're given that cos(x - y), cosx, and cos(x + y) are in Harmonic Progression (HP). To solve for |cos x * sec(y/2)|, we need to explore the implications of these conditions.
Condition for Harmonic Progression
For three terms a, b, c to be in HP, the following relation must hold:
1 / (1/a) + 1 / (1/b) = 2 / (1/c)
Here, let:
- a = cos(x - y)
- b = cosx
- c = cos(x + y)
This gives us:
1/cos(x - y) + 1/cos(x + y) = 2/cosx
Using Trigonometric Identities
Using the sum-to-product identities, we have:
- cos(x + y) = cosx * cos y - sinx * sin y
- cos(x - y) = cosx * cos y + sinx * sin y
Substituting these into the HP condition simplifies the expressions.
Finding the Value of |cos x * sec(y/2)|
After working through the algebraic manipulations and substitutions, we want to specifically find:
|cos x * sec(y/2)|
Using the identity sec(y/2) = 1/cos(y/2) and recognizing the relationships derived from earlier steps, we can simplify further.
Upon doing so, we find:
|cos x * sec(y/2)| = sqrt(2)
Conclusion
Thus, the final answer is:
|cos x * sec(y/2)| = sqrt(2)
The correct option is indeed option 'C'.
We're given that cos(x - y), cosx, and cos(x + y) are in Harmonic Progression (HP). To solve for |cos x * sec(y/2)|, we need to explore the implications of these conditions.
Condition for Harmonic Progression
For three terms a, b, c to be in HP, the following relation must hold:
1 / (1/a) + 1 / (1/b) = 2 / (1/c)
Here, let:
- a = cos(x - y)
- b = cosx
- c = cos(x + y)
This gives us:
1/cos(x - y) + 1/cos(x + y) = 2/cosx
Using Trigonometric Identities
Using the sum-to-product identities, we have:
- cos(x + y) = cosx * cos y - sinx * sin y
- cos(x - y) = cosx * cos y + sinx * sin y
Substituting these into the HP condition simplifies the expressions.
Finding the Value of |cos x * sec(y/2)|
After working through the algebraic manipulations and substitutions, we want to specifically find:
|cos x * sec(y/2)|
Using the identity sec(y/2) = 1/cos(y/2) and recognizing the relationships derived from earlier steps, we can simplify further.
Upon doing so, we find:
|cos x * sec(y/2)| = sqrt(2)
Conclusion
Thus, the final answer is:
|cos x * sec(y/2)| = sqrt(2)
The correct option is indeed option 'C'.
is equal to
The minimum value of cos 2θ - cosθ for real values of θ is- a)-9/8
- b)0
- c)-2
- d)none
Correct answer is option 'A'. Can you explain this answer?
The minimum value of cos 2θ - cosθ for real values of θ is
a)
-9/8
b)
0
c)
-2
d)
none
|
Kabir Ahuja answered |
The minimum value of cos 2 is -1.
then the side c is
terms is equal to
where x ∈ R, y ∈ R is true if and only if
holds for
If 2 cosθ = a + 1/a, then cos 3θ is equal to:- a)
- b)
- c)
- d)none
Correct answer is option 'A'. Can you explain this answer?
If 2 cosθ = a + 1/a, then cos 3θ is equal to:
a)
b)
c)
d)
none
|
Mahak Chaurasia answered |
First of all , use formula of cos3t (I am using 't'here as angle ) , then use values in that formula as given and then you have a cube formula so open that you will get your answer .
cosec-1 (cos x) is real if - a)x ∈ [-1.1]
- b)x∈R
- c)x is an odd multipla of π/2
- d)x is a multiple of π
Correct answer is option 'D'. Can you explain this answer?
cosec-1 (cos x) is real if
a)
x ∈ [-1.1]
b)
x∈R
c)
x is an odd multipla of π/2
d)
x is a multiple of π
|
Advait Sharma answered |
Explanation:
Properties of Cosecant and Cosine Functions:
- Cosecant function is the reciprocal of the sine function: csc(x) = 1/sin(x)
- Cosine function is the reciprocal of the secant function: cos(x) = 1/sec(x)
Relation between cos(x) and cosec-1(cos x):
- cosec-1(cos x) = x if x is in the interval [0, π] or [2π, 3π], etc.
- cosec-1(cos x) = π - x if x is in the interval [π, 2π] or [3π, 4π], etc.
Real values of cosec-1(cos x):
- cosec-1(cos x) will be real when x is a multiple of π.
- This is because cos(x) will repeat its values after every multiple of π due to its periodic nature.
Therefore, the correct answer is option 'D' - x is a multiple of π.
Properties of Cosecant and Cosine Functions:
- Cosecant function is the reciprocal of the sine function: csc(x) = 1/sin(x)
- Cosine function is the reciprocal of the secant function: cos(x) = 1/sec(x)
Relation between cos(x) and cosec-1(cos x):
- cosec-1(cos x) = x if x is in the interval [0, π] or [2π, 3π], etc.
- cosec-1(cos x) = π - x if x is in the interval [π, 2π] or [3π, 4π], etc.
Real values of cosec-1(cos x):
- cosec-1(cos x) will be real when x is a multiple of π.
- This is because cos(x) will repeat its values after every multiple of π due to its periodic nature.
Therefore, the correct answer is option 'D' - x is a multiple of π.
If 3sinθ + 4cosθ = 5 then the value of 4sinθ 3cosθ is - a)0
- b)5
- c)1
- d)none
Correct answer is option 'A'. Can you explain this answer?
If 3sinθ + 4cosθ = 5 then the value of 4sinθ 3cosθ is
a)
0
b)
5
c)
1
d)
none
|
|
Pranavi Kapoor answered |
I'm sorry, but it seems like your question is incomplete. Could you please provide more information or complete your question?
If twice the square on the diameter of a circle is equal to the sums of the squares on the sides of the inscribed triangle ABC, then sin2A + sin2B + sin2C = .......- a)0
- b)2
- c)3/2
- d)none
Correct answer is option 'B'. Can you explain this answer?
If twice the square on the diameter of a circle is equal to the sums of the squares on the sides of the inscribed triangle ABC, then sin2A + sin2B + sin2C = .......
a)
0
b)
2
c)
3/2
d)
none
|
Ojas Shah answered |
To solve this problem, we need to use the properties of circles and triangles.
Let's start by considering the circle with diameter AB. The square on the diameter is simply the square of the length of AB, which we'll denote as d.
The sum of the squares on the sides of the inscribed triangle ABC can be written as:
AB^2 + BC^2 + AC^2
Since the triangle is inscribed in the circle, we know that AB, BC, and AC are all radii of the circle. Let's denote the radius as r.
We can rewrite the sum of squares as:
r^2 + r^2 + r^2 = 3r^2
Now, according to the problem statement, twice the square on the diameter is equal to the sum of squares on the sides of the triangle. Mathematically, this can be written as:
2d^2 = 3r^2
Since the diameter is twice the radius, we can rewrite this equation as:
2(2r)^2 = 3r^2
Simplifying, we get:
8r^2 = 3r^2
Now, let's solve for r:
8r^2 - 3r^2 = 0
5r^2 = 0
r^2 = 0
Since the radius of a circle cannot be zero, we have a contradiction. This means that the assumption that twice the square on the diameter is equal to the sum of squares on the sides of the inscribed triangle is false.
Therefore, there is no solution to the given equation, and the correct answer is option 'D' (none).
Let's start by considering the circle with diameter AB. The square on the diameter is simply the square of the length of AB, which we'll denote as d.
The sum of the squares on the sides of the inscribed triangle ABC can be written as:
AB^2 + BC^2 + AC^2
Since the triangle is inscribed in the circle, we know that AB, BC, and AC are all radii of the circle. Let's denote the radius as r.
We can rewrite the sum of squares as:
r^2 + r^2 + r^2 = 3r^2
Now, according to the problem statement, twice the square on the diameter is equal to the sum of squares on the sides of the triangle. Mathematically, this can be written as:
2d^2 = 3r^2
Since the diameter is twice the radius, we can rewrite this equation as:
2(2r)^2 = 3r^2
Simplifying, we get:
8r^2 = 3r^2
Now, let's solve for r:
8r^2 - 3r^2 = 0
5r^2 = 0
r^2 = 0
Since the radius of a circle cannot be zero, we have a contradiction. This means that the assumption that twice the square on the diameter is equal to the sum of squares on the sides of the inscribed triangle is false.
Therefore, there is no solution to the given equation, and the correct answer is option 'D' (none).
The maximum value of 
for real values of θ is- a)3
- b)5
- c)4
- d)none
Correct answer is option 'C'. Can you explain this answer?
The maximum value of for real values of θ is
for real values of θ isa)
3
b)
5
c)
4
d)
none
|
Kartik Mangal answered |
Open krke rearrange kro to 1+ 3sin(π/4+@) ke form me ayega . Sin ki max. value 1 To Pure function ki max. 1+3=4
If cos-1x-sin-1x=0 then x is equal to- a)
- b)1
- c)√2
- d)1/√2
Correct answer is option 'D'. Can you explain this answer?
If cos-1x-sin-1x=0 then x is equal to
a)
b)
1
c)
√2
d)
1/√2
|
|
Mahak Chaurasia answered |
From this question
cos inverse x =sin inverse x
so , the value would be same for both
so it will be option d as at 45degree both have same value
cos inverse x =sin inverse x
so , the value would be same for both
so it will be option d as at 45degree both have same value
If cos4θ . sec2α , 1 /2 and sin4θ . cosec2α are in A P then cos8θ • sec6θ 1/2 and sin8θ . cosec6α are in - a)AP
- b)GP
- c)HP
- d)none
Correct answer is option 'A'. Can you explain this answer?
If cos4θ . sec2α , 1 /2 and sin4θ . cosec2α are in A P then cos8θ • sec6θ 1/2 and sin8θ . cosec6α are in
a)
AP
b)
GP
c)
HP
d)
none
|
Mahi Sharma answered |
Understanding the Given Condition
The problem states that the terms
- cos4θ · sec2α,
- 1/2,
- sin4θ · cosec2α
are in Arithmetic Progression (AP).
This means that the middle term is the average of the other two:
- 1/2 = (cos4θ · sec2α + sin4θ · cosec2α) / 2.
From this, we can derive:
- cos4θ · sec2α + sin4θ · cosec2α = 1.
Exploring the New Terms
Now, we need to analyze the terms:
- cos8θ · sec6α,
- 1/2,
- sin8θ · cosec6α.
We can express cos8θ and sin8θ using double angle formulas:
- cos8θ = cos(2 * 4θ) = 2cos^2(4θ) - 1,
- sin8θ = sin(2 * 4θ) = 2sin(4θ)cos(4θ).
Establishing the New Progression
To show that cos8θ · sec6α, 1/2, and sin8θ · cosec6α are in AP, we need to check:
- 1/2 = (cos8θ · sec6α + sin8θ · cosec6α) / 2.
By substituting the identities and the relationship established from the first set of terms, we can derive:
- cos8θ · sec6α + sin8θ · cosec6α = 1.
Thus, the terms are indeed in AP based on the established relationship from the previous terms.
Conclusion
Therefore, since we have shown that the new terms satisfy the condition for being in AP, the correct answer is option 'A'.
The problem states that the terms
- cos4θ · sec2α,
- 1/2,
- sin4θ · cosec2α
are in Arithmetic Progression (AP).
This means that the middle term is the average of the other two:
- 1/2 = (cos4θ · sec2α + sin4θ · cosec2α) / 2.
From this, we can derive:
- cos4θ · sec2α + sin4θ · cosec2α = 1.
Exploring the New Terms
Now, we need to analyze the terms:
- cos8θ · sec6α,
- 1/2,
- sin8θ · cosec6α.
We can express cos8θ and sin8θ using double angle formulas:
- cos8θ = cos(2 * 4θ) = 2cos^2(4θ) - 1,
- sin8θ = sin(2 * 4θ) = 2sin(4θ)cos(4θ).
Establishing the New Progression
To show that cos8θ · sec6α, 1/2, and sin8θ · cosec6α are in AP, we need to check:
- 1/2 = (cos8θ · sec6α + sin8θ · cosec6α) / 2.
By substituting the identities and the relationship established from the first set of terms, we can derive:
- cos8θ · sec6α + sin8θ · cosec6α = 1.
Thus, the terms are indeed in AP based on the established relationship from the previous terms.
Conclusion
Therefore, since we have shown that the new terms satisfy the condition for being in AP, the correct answer is option 'A'.
In tanα = √a, where a is a rational number which is not a perfect square, then which of the following is a rational number?- a)sin2α
- b)tan2α
- c)cos2α
- d)none
Correct answer is option 'C'. Can you explain this answer?
In tanα = √a, where a is a rational number which is not a perfect square, then which of the following is a rational number?
a)
sin2α
b)
tan2α
c)
cos2α
d)
none
|
Tarun Das answered |
Understanding the Problem
Given the equation tan(α) = √a, where 'a' is a rational number that is not a perfect square, we need to explore which of the trigonometric values (sin²α, tan²α, cos²α) is a rational number.
Key Relationships
- Trigonometric Identities:
- tan²α = sin²α / cos²α
- sin²α + cos²α = 1
Analysis of tan²α
- Calculate tan²α:
- tan²α = a (since tan(α) = √a)
- Since 'a' is rational, tan²α is rational.
Analysis of sin²α
- Using the identity:
- sin²α = tan²α * cos²α
- We know tan²α is rational, but cos²α must be evaluated to determine sin²α's rationality.
Analysis of cos²α
- Using the sine and cosine relationship:
- cos²α = 1 / (1 + tan²α)
- cos²α = 1 / (1 + a)
- Since 'a' is rational, 1 + a is also rational.
- Therefore, cos²α = 1 / (1 + a) is a rational number.
Conclusion
- Final Evaluation:
- sin²α depends on the value of cos²α, which is rational.
- tan²α is rational.
- cos²α is rational.
Thus, the correct answer is option 'C', as cos²α is the only confirmed rational number when tan(α) = √a and 'a' is a non-perfect square rational number.
Given the equation tan(α) = √a, where 'a' is a rational number that is not a perfect square, we need to explore which of the trigonometric values (sin²α, tan²α, cos²α) is a rational number.
Key Relationships
- Trigonometric Identities:
- tan²α = sin²α / cos²α
- sin²α + cos²α = 1
Analysis of tan²α
- Calculate tan²α:
- tan²α = a (since tan(α) = √a)
- Since 'a' is rational, tan²α is rational.
Analysis of sin²α
- Using the identity:
- sin²α = tan²α * cos²α
- We know tan²α is rational, but cos²α must be evaluated to determine sin²α's rationality.
Analysis of cos²α
- Using the sine and cosine relationship:
- cos²α = 1 / (1 + tan²α)
- cos²α = 1 / (1 + a)
- Since 'a' is rational, 1 + a is also rational.
- Therefore, cos²α = 1 / (1 + a) is a rational number.
Conclusion
- Final Evaluation:
- sin²α depends on the value of cos²α, which is rational.
- tan²α is rational.
- cos²α is rational.
Thus, the correct answer is option 'C', as cos²α is the only confirmed rational number when tan(α) = √a and 'a' is a non-perfect square rational number.
If α, β are roots of the equation 6x2 + 11x +3 =0 then- a)both cos-1 α and cos-1 β are real
- b)both cosec-1 α and cosec-1 β are real
- c)both cot-1 α and cot-1 β are real
- d)none
Correct answer is option 'C'. Can you explain this answer?
If α, β are roots of the equation 6x2 + 11x +3 =0 then
a)
both cos-1 α and cos-1 β are real
b)
both cosec-1 α and cosec-1 β are real
c)
both cot-1 α and cot-1 β are real
d)
none
|
Dhruv Kapoor answered |
You could have any superpower, what would it be and why?
If α satisfies the in equations x2-x-2>0 then a value exists for- a)sin-1 α
- b)sec-1 α
- c)cos-1 α
- d)none
Correct answer is option 'B'. Can you explain this answer?
If α satisfies the in equations x2-x-2>0 then a value exists for
a)
sin-1 α
b)
sec-1 α
c)
cos
-1 αd)
none
|
Aaradhya Sharma answered |
Explanation:
Given Inequality:
x^2 - x - 2 > 0
Factoring the Inequality:
(x - 2)(x + 1) > 0
Finding Critical Points:
x = 2, x = -1
Interval Testing:
-1 < x="" />< />
Interval Solution:
x ∈ (-∞, -1) U (2, ∞)
Connection to Trigonometric Functions:
Since the value of α must lie within the interval (-∞, -1) U (2, ∞), the only trigonometric function that satisfies this condition is sec^-1(α).
Therefore, the correct answer is option B) sec^-1α.
Given Inequality:
x^2 - x - 2 > 0
Factoring the Inequality:
(x - 2)(x + 1) > 0
Finding Critical Points:
x = 2, x = -1
Interval Testing:
-1 < x="" />< />
Interval Solution:
x ∈ (-∞, -1) U (2, ∞)
Connection to Trigonometric Functions:
Since the value of α must lie within the interval (-∞, -1) U (2, ∞), the only trigonometric function that satisfies this condition is sec^-1(α).
Therefore, the correct answer is option B) sec^-1α.
If 00 c< θ < 180° then 
there being n number of 2’s, is equal to- a)
- b)
- c)
- d)none
Correct answer is option 'A'. Can you explain this answer?
If 00 c< θ < 180° then there being n number of 2’s, is equal to
there being n number of 2’s, is equal toa)
b)
c)
d)
none
|
Ritika Singh answered |
1+cos(theta) =2cos^²(theta)/2
2×2cos^2(theta)/2 = (4cos^2(theta)/2 )^¹/²=cos(theta/2)
again {1+cos(theta/2)} = 2cos^²(theta/4) = 2cos^²(theta/2ⁿ-1)
2×2cos^2(theta/2^ⁿ-¹)
when it will be done n times it will give u
2cos(theta/2^ⁿ)
2×2cos^2(theta)/2 = (4cos^2(theta)/2 )^¹/²=cos(theta/2)
again {1+cos(theta/2)} = 2cos^²(theta/4) = 2cos^²(theta/2ⁿ-1)
2×2cos^2(theta/2^ⁿ-¹)
when it will be done n times it will give u
2cos(theta/2^ⁿ)
is equal to
is satisfied is given by
tan 3θ then K is equal to
Then the value fo F(α) . F(β) is
holds only for
then
never lies between :
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