Solving the Trigonometry Problem in a Right Triangle

We are given a right triangle \(\Delta ABC\) where \(\angle B = 90^\circ\). We are also given that \(\tan A = \frac{1}{\sqrt{2}}\). We need to find the value of the expression \(\sin A \cos C + \cos A \sin C\).

Understanding the Given Information

• \(\Delta ABC\) is a right triangle.
• \(\angle B = 90^\circ\). This is the right angle.
• \(\tan A = \frac{1}{\sqrt{2}}\). This relates the sides opposite and adjacent to angle A.
• We need to evaluate \(\sin A \cos C + \cos A \sin C\).

Method 1: Using Trigonometric Identities and Triangle Angle Properties

Let's look at the expression we need to evaluate: \(\sin A \cos C + \cos A \sin C\).

This expression is the standard expansion for the sine of the sum of two angles, i.e., \(\sin(A+C)\).

So, \(\sin A \cos C + \cos A \sin C = \sin(A+C)\).

Now, let's consider the angles in triangle \(\Delta ABC\). The sum of angles in any triangle is \(180^\circ\). So, for \(\Delta ABC\), we have:

\(A + B + C = 180^\circ\)

We are given that \(\angle B = 90^\circ\). Substituting this into the equation:

\(A + 90^\circ + C = 180^\circ\)

Subtracting \(90^\circ\) from both sides, we get:

\(A + C = 180^\circ - 90^\circ\)

\(A + C = 90^\circ\)

This means that angles A and C are complementary angles in this right triangle.

Now, we can substitute this value of \((A+C)\) back into the expression \(\sin(A+C)\):

Value = \(\sin(90^\circ)\)

The sine of \(90^\circ\) is a standard trigonometric value:

\(\sin(90^\circ) = 1\)

Therefore, the value of \(\sin A \cos C + \cos A \sin C\) is 1.

Notice that in this method, the information about \(\tan A = \frac{1}{\sqrt{2}}\) was not strictly needed to find the value of the expression, as it simplified based on the properties of a right triangle and trigonometric identities.

Method 2: Using the Given \(\tan A\) Value to Find Sides and Ratios

We are given \(\tan A = \frac{1}{\sqrt{2}}\) in the right triangle \(\Delta ABC\) where \(\angle B = 90^\circ\). We know that \(\tan A = \frac{\text{Opposite side to A}}{\text{Adjacent side to A}}\).

Let the side opposite to angle A be BC, and the side adjacent to angle A (excluding the hypotenuse) be AB. Let the hypotenuse be AC.

So, \(\tan A = \frac{BC}{AB} = \frac{1}{\sqrt{2}}\).

We can let \(BC = 1k\) and \(AB = \sqrt{2}k\) for some positive constant \(k\). For simplicity, we can just use \(BC = 1\) and \(AB = \sqrt{2}\).

Now, we can find the hypotenuse AC using the Pythagorean theorem:

\(AC^2 = AB^2 + BC^2\)

\(AC^2 = (\sqrt{2})^2 + (1)^2\)

\(AC^2 = 2 + 1\)

\(AC^2 = 3\)

\(AC = \sqrt{3}\) (Since length must be positive)

Now we can find the sine and cosine of angle A:

• \(\sin A = \frac{\text{Opposite side to A}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{1}{\sqrt{3}}\)
• \(\cos A = \frac{\text{Adjacent side to A}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{\sqrt{2}}{\sqrt{3}}\)

Next, let's find the sine and cosine of angle C. In a right triangle, angles A and C are complementary, meaning \(A + C = 90^\circ\). This implies:

• \(\sin C = \sin(90^\circ - A) = \cos A = \frac{\sqrt{2}}{\sqrt{3}}\)
• \(\cos C = \cos(90^\circ - A) = \sin A = \frac{1}{\sqrt{3}}\)

Alternatively, from angle C's perspective:

• The side opposite to C is AB, with length \(\sqrt{2}\).
• The side adjacent to C is BC, with length 1.
• The hypotenuse is AC, with length \(\sqrt{3}\).

• \(\sin C = \frac{\text{Opposite side to C}}{\text{Hypotenuse}} = \frac{AB}{AC} = \frac{\sqrt{2}}{\sqrt{3}}\)
• \(\cos C = \frac{\text{Adjacent side to C}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{1}{\sqrt{3}}\)

These values match what we got using the complementary angle relationships.

Now, substitute these values into the expression \(\sin A \cos C + \cos A \sin C\):

\(\sin A \cos C + \cos A \sin C = \left(\frac{1}{\sqrt{3}}\right) \left(\frac{1}{\sqrt{3}}\right) + \left(\frac{\sqrt{2}}{\sqrt{3}}\right) \left(\frac{\sqrt{2}}{\sqrt{3}}\right)\)

\(= \frac{1}{\sqrt{3} \times \sqrt{3}} + \frac{\sqrt{2} \times \sqrt{2}}{\sqrt{3} \times \sqrt{3}}\)

\(= \frac{1}{3} + \frac{2}{3}\)

\(= \frac{1+2}{3}\)

\(= \frac{3}{3}\)

\(= 1\)

Both methods give the same result, confirming the answer.

Conclusion

The value of \(\sin A \cos C + \cos A \sin C\) in the given right triangle \(\Delta ABC\) with \(\angle B = 90^\circ\) is 1. This is because the expression simplifies to \(\sin(A+C)\), and for a right triangle with angle B = \(90^\circ\), the sum of the other two angles A and C must be \(90^\circ\).

Revision Table: Key Trigonometry Concepts

Additional Information on Right Triangle Trigonometry

Trigonometry is the study of the relationships between the angles and sides of triangles. Right triangles are fundamental in trigonometry because the trigonometric ratios (sine, cosine, tangent) are defined based on the ratios of the sides relative to the acute angles in a right triangle.

• SOH CAH TOA: This is a mnemonic to remember the definitions of the basic trigonometric ratios:
  • \(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
  • \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
  • \(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}\)
• Pythagorean Theorem: In a right triangle with legs a and b and hypotenuse c, the relationship \(a^2 + b^2 = c^2\) holds true. This is crucial for finding unknown side lengths.
• Angle Sum Property: The fact that angles in a triangle sum to \(180^\circ\) allows us to find unknown angles or establish relationships between them, like the complementary nature of the acute angles in a right triangle.
• Trigonometric Identities: These are equations involving trigonometric functions that are true for all values of the variables. The sum identities, like \(\sin(A+C) = \sin A \cos C + \cos A \sin C\), are very useful for simplifying expressions or solving equations.

Understanding these basic concepts is key to solving problems involving right triangles and trigonometry, like the one discussed here. The problem beautifully illustrates how combining the angle properties of a right triangle with trigonometric identities can lead to a quick solution, even when specific side ratios are given.