https://file.aiquickdraw.com/mathbot/file/1727780356834a52r50hh.jpg

The expression given is:

\[
\frac{\sin 90^\circ}{\cos 0^\circ} - \frac{\sin 60^\circ}{\sec 30^\circ} + \frac{\tan 30^\circ}{\sec 30^\circ}
\]

Let's break it down and evaluate step-by-step:

### Step 1: Evaluate Trigonometric Values
- \(\sin 90^\circ = 1\)
- \(\cos 0^\circ = 1\)
- \(\sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}\)
- \(\tan 30^\circ = \frac{1}{\sqrt{3}}\)

### Step 2: Substitute These Values
Now we substitute the trigonometric values into the expression:

\[
\frac{1}{1} - \frac{\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}} + \frac{\frac{1}{\sqrt{3}}}{\frac{2}{\sqrt{3}}}
\]

### Step 3: Simplify Each Term
- First term: \(\frac{1}{1} = 1\)
- Second term: \(\frac{\frac{\sqrt{3}}{2}}{\frac{2}{\sqrt{3}}} = \frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4}\)
- Third term: \(\frac{\frac{1}{\sqrt{3}}}{\frac{2}{\sqrt{3}}} = \frac{1}{2}\)

### Step 4: Final Calculation
Now we combine everything:

\[
1 - \frac{3}{4} + \frac{1}{2}
\]

First, subtract \(\frac{3}{4}\) from 1:

\[
1 - \frac{3}{4} = \frac{4}{4} - \frac{3}{4} = \frac{1}{4}
\]

Now add \(\frac{1}{2}\):

\[
\frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{3}{4}
\]

### Final Answer:
The value of the given expression is:

\[
\frac{3}{4}
\]

Would you like a further breakdown of any steps or details on the trigonometric functions used?

Here are five questions to expand your understanding:
1. What are the fundamental trigonometric identities that simplify expressions like these?
2. How do we convert between radians and degrees in trigonometry?
3. Can you explain how the secant function relates to the cosine function?
4. What are the steps to rationalize a trigonometric fraction?
5. How would this expression change if any angle were different, say \( \sin 45^\circ \) instead of \( \sin 60^\circ \)?

**Tip:** Remember that secant and tangent are reciprocal of cosine and sine, respectively, which simplifies many trigonometric expressions easily.

Simplify the expression: (sin(90°)/cos(0°)) - (sin(60°)/sec(30°)) + (tan(30°)/sec(30°))

Learn how to simplify the expression involving sin(90°), cos(0°), sin(60°), sec(30°), and tan(30°) step-by-step. This problem covers key trigonometric identities and is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation