We are given a right triangle with one angle measuring $46^\circ$ and the hypotenuse measuring 28 units. We need to find the missing angle $A$, the side $a$, and the side $b$. Let's break it down:

Step 1: Find angle $A$

The sum of the angles in a triangle is $180^\circ$. Since this is a right triangle, one angle is $90^\circ$. So, the other angle $A$ can be found by subtracting the known angles from $180^\circ$: $A = 90^\circ - 46^\circ = 44^\circ$

Step 2: Use trigonometry to find sides $a$ and $b$

We know the angle $46^\circ$ and the hypotenuse, so we can use the sine and cosine functions:

• For side $a$ (opposite to $46^\circ$): $\sin(46^\circ) = \frac{a}{28}$ Solving for $a$: $a = 28 \times \sin(46^\circ) \approx 28 \times 0.7193 \approx 20.1$

• For side $b$ (adjacent to $46^\circ$): $\cos(46^\circ) = \frac{b}{28}$ Solving for $b$: $b = 28 \times \cos(46^\circ) \approx 28 \times 0.6947 \approx 19.5$

Final answers:

• $A = 44^\circ$
• $a \approx 20.1$
• $b \approx 19.5$

Would you like more details or have any other questions?

Here are 5 related questions:

1. How can we use the tangent function to verify the sides?
2. What would happen if the angle was different but the hypotenuse stayed the same?
3. How do we solve for angles if two sides are given?
4. What is the Pythagorean Theorem, and how does it apply here?
5. Can we solve this triangle without trigonometric functions?

Tip: Always check if a triangle problem gives you a right angle; this helps identify which trigonometric ratios to use easily.