Find the value of :
`3sin^(2)30^(@)+2tan^(2)60^(@)-5cos^(2)45^(@)`

Answer

To solve the expression \(3\sin^2(30^\circ) + 2\tan^2(60^\circ) - 5\cos^2(45^\circ)\), we will follow these steps:

### Step 1: Identify the values of the trigonometric functions
- We know the standard values:
- \(\sin(30^\circ) = \frac{1}{2}\)
- \(\tan(60^\circ) = \sqrt{3}\)
- \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)

### Step 2: Substitute the values into the expression
Now we substitute these values into the expression:
\[ 3\sin^2(30^\circ) + 2\tan^2(60^\circ) - 5\cos^2(45^\circ) = 3\left(\frac{1}{2}\right)^2 + 2\left(\sqrt{3}\right)^2 - 5\left(\frac{1}{\sqrt{2}}\right)^2 \]

### Step 3: Calculate the squares
Calculating the squares:
- \(\sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
- \(\tan^2(60^\circ) = \left(\sqrt{3}\right)^2 = 3\)
- \(\cos^2(45^\circ) = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2}\)

### Step 4: Substitute the squared values back into the expression
Now substitute these squared values back into the expression:
\[ 3\left(\frac{1}{4}\right) + 2(3) - 5\left(\frac{1}{2}\right) \]

### Step 5: Simplify each term
Now simplify each term:
- \(3 \cdot \frac{1}{4} = \frac{3}{4}\)
- \(2 \cdot 3 = 6\)
- \(5 \cdot \frac{1}{2} = \frac{5}{2}\)

### Step 6: Rewrite the expression
The expression now looks like this:
\[ \frac{3}{4} + 6 - \frac{5}{2} \]

### Step 7: Convert all terms to have a common denominator
The common denominator for \(4\), \(1\) (for \(6\)), and \(2\) is \(4\):
- \(6 = \frac{24}{4}\)
- \(-\frac{5}{2} = -\frac{10}{4}\)

### Step 8: Combine the terms
Now combine the terms:
\[ \frac{3}{4} + \frac{24}{4} - \frac{10}{4} = \frac{3 + 24 - 10}{4} = \frac{17}{4} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{17}{4} \]

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