        
(a) Without using Trigonometric tables, find the value of 
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�𝑖𝑛2 30° −2 𝑐𝑜𝑠3 60°+3 𝑡𝑎𝑛4 45° 
(b)

To solve the expression without using trigonometric tables, we must rely on standard known values for the sine, cosine, and tangent of common angles (like 30°, 60°, and 45°). Let’s break it down step by step:

### Given expression:
\[
\sin^2 30^\circ - 2\cos^3 60^\circ + 3\tan^4 45^\circ
\]

### Step 1: Evaluate \(\sin 30^\circ\)
It is well known that:
\[
\sin 30^\circ = \frac{1}{2}
\]
Thus,
\[
\sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]

### Step 2: Evaluate \(\cos 60^\circ\)
Similarly, we know:
\[
\cos 60^\circ = \frac{1}{2}
\]
Thus,
\[
\cos^3 60^\circ = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]

### Step 3: Evaluate \(\tan 45^\circ\)
\[
\tan 45^\circ = 1
\]
Thus,
\[
\tan^4 45^\circ = 1^4 = 1
\]

### Step 4: Substitute the values into the original expression:
\[
\sin^2 30^\circ - 2\cos^3 60^\circ + 3\tan^4 45^\circ
\]
Substituting the values we calculated:
\[
= \frac{1}{4} - 2\left(\frac{1}{8}\right) + 3(1)
\]
Simplifying:
\[
= \frac{1}{4} - \frac{2}{8} + 3 = \frac{1}{4} - \frac{1}{4} + 3
\]

### Step 5: Simplify the result:
\[
\frac{1}{4} - \frac{1}{4} + 3 = 0 + 3 = 3
\]

### Final Answer:
\[
\boxed{3}
\]

Would you like further details or clarifications on any part of this solution?

---

Here are 5 related questions for practice:
1. What is the value of \(\sin^2 45^\circ + \cos^2 45^\circ\)?
2. Simplify the expression: \(\sin^2 60^\circ + \cos^2 30^\circ\).
3. What is \(\tan 60^\circ\) and how does it compare to \(\sin 60^\circ\)?
4. Prove that \(\sin^2 x + \cos^2 x = 1\) for any angle \(x\).
5. Evaluate \(\sin 45^\circ \cdot \cos 45^\circ\).

**Tip:** Always remember basic trigonometric values for standard angles (0°, 30°, 45°, 60°, and 90°) for quick reference in problems like this.

(a) Without using Trigonometric tables, find the value of sin²(30°) − 2 cos³(60°) + 3 tan⁴(45°). (b)

Solve the trigonometric expression sin²(30°) − 2 cos³(60°) + 3 tan⁴(45°) without using trigonometric tables. Follow the step-by-step breakdown of basic trigonometric values, powers, and simplification. Suitable for high school students in Grades 9-12.

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