Solving Trigonometric Expression in a Right-Angled Triangle

The question asks us to find the value of a trigonometric expression, $\sin^2\theta - \cos^2\theta$, in a right-angled triangle ABC, where the right angle is at B, and we are given the lengths of two sides, AB and AC. The angle $\theta$ is given as $\angle ACB$.

In a right-angled triangle ABC, with the right angle at B:

• Hypotenuse (the side opposite the right angle) = AC = 10 units.
• One leg = AB = 8 units.
• The other leg = BC = ?
• The angle $\theta = \angle ACB$ is the angle at vertex C.

To find the trigonometric ratios for angle $\theta$, we first need to find the length of the side BC, which is adjacent to angle $\theta$. We can use the Pythagorean theorem for this.

Using the Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

In $\triangle ABC$, we have:

\begin{equation*} AB^2 + BC^2 = AC^2 \end{equation*}

Substitute the given values:

\begin{equation*} 8^2 + BC^2 = 10^2 \end{equation*}

\begin{equation*} 64 + BC^2 = 100 \end{equation*}

Subtract 64 from both sides:

\begin{equation*} BC^2 = 100 - 64 \end{equation*}

\begin{equation*} BC^2 = 36 \end{equation*}

Take the square root of both sides:

\begin{equation*} BC = \sqrt{36} \end{equation*}

\begin{equation*} BC = 6 \text{ units} \end{equation*}

So, the lengths of the sides of the triangle are AB = 8, BC = 6, and AC = 10.

Calculating Sine and Cosine of Angle $\theta$

For angle $\theta = \angle ACB$ in the right-angled triangle ABC:

• The side opposite to $\theta$ is AB = 8.
• The side adjacent to $\theta$ is BC = 6.
• The hypotenuse is AC = 10.

The sine of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.

\begin{equation*} \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} \end{equation*}

Substitute the values:

\begin{equation*} \sin \theta = \frac{8}{10} = \frac{4}{5} \end{equation*}

The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.

\begin{equation*} \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC} \end{equation*}

Substitute the values:

\begin{equation*} \cos \theta = \frac{6}{10} = \frac{3}{5} \end{equation*}

Calculating $\sin^2\theta - \cos^2\theta$

Now we need to find the value of $\sin^2\theta - \cos^2\theta$.

First, calculate $\sin^2\theta$ and $\cos^2\theta$:

\begin{equation*} \sin^2 \theta = (\sin \theta)^2 = \left(\frac{4}{5}\right)^2 = \frac{4^2}{5^2} = \frac{16}{25} \end{equation*}

\begin{equation*} \cos^2 \theta = (\cos \theta)^2 = \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} \end{equation*}

Now, substitute these values into the expression $\sin^2\theta - \cos^2\theta$:

\begin{equation*} \sin^2\theta - \cos^2\theta = \frac{16}{25} - \frac{9}{25} \end{equation*}

Subtract the fractions:

\begin{equation*} \sin^2\theta - \cos^2\theta = \frac{16 - 9}{25} = \frac{7}{25} \end{equation*}

The value of $\sin^2\theta - \cos^2\theta$ is $\frac{7}{25}$.

Comparing with Options

Let's compare our result with the given options:

• Option 1: $\frac{9}{25}$
• Option 2: $\frac{7}{25}$
• Option 3: $\frac{22}{25}$
• Option 4: $\frac{16}{25}$

Our calculated value, $\frac{7}{25}$, matches Option 2.

The final answer is $\frac{7}{25}$.

Revision Table: Right Triangle Trigonometry

Additional Information: Trigonometric Identities

The expression $\cos^2\theta - \sin^2\theta$ is a common trigonometric identity for $\cos(2\theta)$. Our result $\sin^2\theta - \cos^2\theta$ is the negative of this identity, i.e., $-\cos(2\theta)$.

Let's quickly check this. We found $\cos\theta = \frac{3}{5}$ and $\sin\theta = \frac{4}{5}$.

The double angle formula for cosine is:

\begin{equation*} \cos(2\theta) = \cos^2\theta - \sin^2\theta \end{equation*}

Using our values:

\begin{equation*} \cos(2\theta) = \left(\frac{3}{5}\right)^2 - \left(\frac{4}{5}\right)^2 = \frac{9}{25} - \frac{16}{25} = -\frac{7}{25} \end{equation*}

The expression we calculated is $\sin^2\theta - \cos^2\theta$, which is:

\begin{equation*} \sin^2\theta - \cos^2\theta = - (\cos^2\theta - \sin^2\theta) = - \cos(2\theta) \end{equation*}

Since $\cos(2\theta) = -\frac{7}{25}$, our result $\sin^2\theta - \cos^2\theta$ is indeed $- (-\frac{7}{25}) = \frac{7}{25}$. This confirms our calculation using basic definitions.