IDNLearn.com is designed to help you find reliable answers to any question you have. Get step-by-step guidance for all your technical questions from our dedicated community members.
Sagot :
Of course! Let's solve and verify the trigonometric identity step-by-step:
Given identity to verify:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
### Step 1: Simplify the left-hand side
First, we start by expanding the left-hand side of the given equation.
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) \][/tex]
We use the difference of squares formula [tex]\( (a+b)(a-b) = a^2 - b^2 \)[/tex]:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = (2 \cos \theta)^2 - 1^2 \][/tex]
Now, compute each part:
[tex]\[ (2 \cos \theta)^2 = 4 \cos^2 \theta \][/tex]
[tex]\[ 1^2 = 1 \][/tex]
So, substituting back, we have:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 4 \cos^2 \theta - 1 \][/tex]
Thus, the left-hand side simplifies to:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
### Step 2: Simplify the right-hand side
Now let's simplify the right-hand side of the original equation:
[tex]\[ 2 \cos 2\theta + 1 \][/tex]
### Step 3: Use a known trigonometric identity
We will now use a trigonometric identity to relate [tex]\(\cos(2\theta)\)[/tex] directly with [tex]\(\cos(\theta)\)[/tex]. One of the double angle identities for cosine is:
[tex]\[ \cos(2\theta) = 2 \cos^2 \theta - 1 \][/tex]
Therefore, substitute [tex]\(\cos(2\theta)\)[/tex] in the right-hand side expression:
[tex]\[ 2 \cos(2\theta) + 1 = 2(2 \cos^2 \theta - 1) + 1 \][/tex]
Simplify inside the parentheses first:
[tex]\[ = 2 \cdot 2 \cos^2 \theta - 2 \cdot 1 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 2 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 1 \][/tex]
### Step 4: Compare both sides
Now, compare the simplified forms of both sides of the original equation:
Left-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Right-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Since the simplified forms of both sides are equal, we have verified the identity is true.
### Conclusion
Therefore, the identity:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
is verified to be true.
Given identity to verify:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
### Step 1: Simplify the left-hand side
First, we start by expanding the left-hand side of the given equation.
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) \][/tex]
We use the difference of squares formula [tex]\( (a+b)(a-b) = a^2 - b^2 \)[/tex]:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = (2 \cos \theta)^2 - 1^2 \][/tex]
Now, compute each part:
[tex]\[ (2 \cos \theta)^2 = 4 \cos^2 \theta \][/tex]
[tex]\[ 1^2 = 1 \][/tex]
So, substituting back, we have:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 4 \cos^2 \theta - 1 \][/tex]
Thus, the left-hand side simplifies to:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
### Step 2: Simplify the right-hand side
Now let's simplify the right-hand side of the original equation:
[tex]\[ 2 \cos 2\theta + 1 \][/tex]
### Step 3: Use a known trigonometric identity
We will now use a trigonometric identity to relate [tex]\(\cos(2\theta)\)[/tex] directly with [tex]\(\cos(\theta)\)[/tex]. One of the double angle identities for cosine is:
[tex]\[ \cos(2\theta) = 2 \cos^2 \theta - 1 \][/tex]
Therefore, substitute [tex]\(\cos(2\theta)\)[/tex] in the right-hand side expression:
[tex]\[ 2 \cos(2\theta) + 1 = 2(2 \cos^2 \theta - 1) + 1 \][/tex]
Simplify inside the parentheses first:
[tex]\[ = 2 \cdot 2 \cos^2 \theta - 2 \cdot 1 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 2 + 1 \][/tex]
[tex]\[ = 4 \cos^2 \theta - 1 \][/tex]
### Step 4: Compare both sides
Now, compare the simplified forms of both sides of the original equation:
Left-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Right-hand side:
[tex]\[ 4 \cos^2 \theta - 1 \][/tex]
Since the simplified forms of both sides are equal, we have verified the identity is true.
### Conclusion
Therefore, the identity:
[tex]\[ (2 \cos \theta + 1)(2 \cos \theta - 1) = 2 \cos 2\theta + 1 \][/tex]
is verified to be true.
For the first one 3x= 6x-2 the answer is 1.5 since the 6- is positive you want to subtract it with 3x since it’s the only other number with x, you subtract 3x-6x and get -3x, now the equation is -3x=-2, Divide the -2 with -3x to get rid of the -2, -2/-2, -3x/-2
X=1.5
X=1.5
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
