Which of the following are trigonometric identities?
(Can be more then one answer)
tanx cosx cscx = 1
secx-cosx/secs=sin^2x
1-tanxtany=cos(x+y)/cosxcosy
4cosx sinx = 2cosx + 1 - 2sinx
Find all solutions to the equation cosx cos(3x) - sinx sin(3x) = 0 on the interval [0,2π]. (Points : 5)
π/8,3π/8,5π/8,7π/8,9π/8,11π/8,13π/8,15π/8
π/8,5π/8,9π/8,13π/8
3π/8,7π/8,11π/8,15π/8
π/8,9π/8, 15π/8
tanx cosx cscx
= sinx/cosx * cosx * 1/sinx
= 1
so the first one is an identity
For questions like this, I use this trick
pick a weird angle, e.g x = 13.79°
If that weird angle satisfies your equation, it is "highly likely" that it is an identify.
( I know that it does not prove it, but I would put a large sum of money on it)
2nd one works
3rd works
4th does not
cosx cos(3x) - sinx sin(3x0 = 0
You should recognize that pattern and get
cos(x+3x) = 0
cos (4x) = 0
4x = π/2 or 4x = 3π/2
x = π/8 or x = 3π/8
now the period of cos (4x) = 2π/4 or π/2
so by adding multiples of π/2 to any answer will give up new answers
π/8
π/8 + π/2 = 5π/8
5π/8 + π/2 = 9π/8
9π/8 + π/2 = 13π/8
etc. (numerator jumping by 4π)
same for 3π/8 , adding π/2 yields more answers
3π/8 , 7π/8, 11π/8 , 15π/8 , next one would be > 2π
looks like you answer #1, with 8 different answers.
The trigonometric identities are:
1. tanx cosx cscx = 1
2. secx - cosx/secx = sin^2x
3. 1 - tanx tany = cos(x+y)/cosx cosy
Out of these options, only the first identity (tanx cosx cscx = 1) is a trigonometric identity.
To find the solutions to the equation cosx cos(3x) - sinx sin(3x) = 0 on the interval [0,2π], we can rewrite it as:
cos(x - 3x) = 0
cos(-2x) = 0
Using the double-angle identity for cosine, we have:
cos(-2x) = cos(2x) = 0
The solutions to this equation can be found by setting 2x equal to the angles where cos(2x) = 0. These angles are π/4 and 3π/4. Since we need to find the solutions on the interval [0,2π], we can add 2π to each solution to find all the solutions. Thus, the solutions are:
2x = π/4 + 2πk, where k is an integer
2x = 3π/4 + 2πk, where k is an integer
Dividing both sides by 2, we get:
x = π/8 + πk, where k is an integer
x = 3π/8 + πk, where k is an integer
x = 5π/8 + πk, where k is an integer
x = 7π/8 + πk, where k is an integer
x = 9π/8 + πk, where k is an integer
x = 11π/8 + πk, where k is an integer
x = 13π/8 + πk, where k is an integer
x = 15π/8 + πk, where k is an integer
From these solutions, we can see that the options provided are:
π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, 15π/8
Therefore, the correct answer is:
π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, 15π/8
To determine which of the given equations are trigonometric identities, we need to see if they hold true for all values of x. Let's analyze each equation individually:
1) tanx cosx cscx = 1
This equation is a trigonometric identity because it holds true for all values of x. If we want to verify it, we can rewrite cscx as 1/sinx and then substitute it into the equation. By simplifying the expression, we arrive at the result 1 = 1, confirming that it is indeed an identity.
2) secx - cosx / secx = sin^2x
This equation is not a trigonometric identity because it doesn't hold true for all values of x. We can verify this by selecting specific values of x where it doesn't hold true.
3) 1 - tanx tany = cos(x + y) / cosx cosy
This equation is a trigonometric identity because it holds true for all values of x and y. We can verify this by rewriting tanx and tany as sinx/cosx and siny/cosy, respectively. By simplifying the expression, we can see that both sides are equal.
4) 4cosx sinx = 2cosx + 1 - 2sinx
This equation is not a trigonometric identity because it doesn't hold true for all values of x. We can verify this by selecting specific values of x where it doesn't hold true.
Now, let's move on to finding all the solutions to the equation cosx cos(3x) - sinx sin(3x) = 0 on the interval [0, 2π].
To solve this equation, we can rewrite it using the cosine double-angle identity:
cos(x-y) = cosx cosy + sinx siny
By substituting y = 2x into the equation, we get:
cos(x-2x) = cosx cos(2x) + sinx sin(2x)
cos(-x) = cosx (2cos^2x - 1) + sinx (2sinxcosx)
Simplifying further:
cos(x) = 2cos^3x - cosx + 2sin^2x cosx
Now, we have a cubic equation in terms of cosx. To solve this equation, we can factor out cosx:
cos(x)(1 - 2cos^2x) + 2sin^2x cosx = 0
Setting each factor equal to zero:
cos(x) = 0 ----> Equation A
1 - 2cos^2x = 0 ----> Equation B
Solving Equation A:
cos(x) = 0
x = π/2, 3π/2
Solving Equation B:
1 - 2cos^2x = 0
cos^2x = 1/2
cosx = ±√(1/2)
cosx = ±(1/√2)
cosx = ±(√2/2)
To find the corresponding values of x for the cosine values, we can use the unit circle and determine the angles where cosx = ±(√2/2). These angles are π/4, 3π/4, 5π/4, and 7π/4.
Combining the solutions from both Equation A and Equation B, we get the following solutions for x on the interval [0, 2π]:
π/2, 3π/2, π/4, 3π/4, 5π/4, 7π/4
Hence, the correct answer is: π/8, 3π/8, 5π/8, 7π/8, 9π/8, 11π/8, 13π/8, 15π/8.