Trigonometric Expression Evaluation: Solving for sec 23° - sin 67°

We are given that \( \sin 23^\circ = \frac{a}{b} \) and asked to find the value of \( \sec 23^\circ - \sin 67^\circ \). To solve this, we will use basic trigonometric definitions and complementary angle identities.

Understanding the Given Information

We have the sine of an angle, \( 23^\circ \), as a ratio of two quantities, \( a \) and \( b \). In a right-angled triangle, \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \). So, for an angle of \( 23^\circ \), we can consider the opposite side to be proportional to \( a \) and the hypotenuse to be proportional to \( b \).

Finding Other Trigonometric Ratios for 23°

Using the Pythagorean theorem (\(\text{Opposite}^2 + \text{Adjacent}^2 = \text{Hypotenuse}^2\)), we can find the adjacent side:

\( a^2 + \text{Adjacent}^2 = b^2 \)

\( \text{Adjacent}^2 = b^2 - a^2 \)

\( \text{Adjacent} = \sqrt{b^2 - a^2} \)

Now we can find \( \cos 23^\circ \) and \( \sec 23^\circ \):

• \( \cos 23^\circ = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{\sqrt{b^2 - a^2}}{b} \)
• \( \sec 23^\circ = \frac{1}{\cos 23^\circ} = \frac{b}{\sqrt{b^2 - a^2}} \)

Using Complementary Angle Identity for sin 67°

We need to evaluate \( \sin 67^\circ \). Notice that \( 67^\circ + 23^\circ = 90^\circ \). This means \( 67^\circ \) and \( 23^\circ \) are complementary angles. The complementary angle identity for sine is:

\( \sin (90^\circ - \theta) = \cos \theta \)

Applying this identity with \( \theta = 23^\circ \):

\( \sin 67^\circ = \sin (90^\circ - 23^\circ) = \cos 23^\circ \)

From our previous calculation, we know that \( \cos 23^\circ = \frac{\sqrt{b^2 - a^2}}{b} \). Therefore:

\( \sin 67^\circ = \frac{\sqrt{b^2 - a^2}}{b} \)

Evaluating sec 23° - sin 67°

Now substitute the values we found for \( \sec 23^\circ \) and \( \sin 67^\circ \) into the expression \( \sec 23^\circ - \sin 67^\circ \):

\( \sec 23^\circ - \sin 67^\circ = \frac{b}{\sqrt{b^2 - a^2}} - \frac{\sqrt{b^2 - a^2}}{b} \)

To subtract these fractions, we find a common denominator, which is \( b \sqrt{b^2 - a^2} \):

\( = \frac{b \cdot b}{b \sqrt{b^2 - a^2}} - \frac{\sqrt{b^2 - a^2} \cdot \sqrt{b^2 - a^2}}{b \sqrt{b^2 - a^2}} \)

\( = \frac{b^2 - (b^2 - a^2)}{b \sqrt{b^2 - a^2}} \)

\( = \frac{b^2 - b^2 + a^2}{b \sqrt{b^2 - a^2}} \)

\( = \frac{a^2}{b \sqrt{b^2 - a^2}} \)

Thus, the value of \( \sec 23^\circ - \sin 67^\circ \) is \( \frac{a^2}{b \sqrt{b^2 - a^2}} \).

Conclusion on the Value of sec 23° - sin 67°

Based on our calculations, the value of \( \sec 23^\circ - \sin 67^\circ \) is \( \frac{a^2}{b \sqrt{b^2 - a^2}} \). This matches one of the provided options.

Revision Table: Key Trigonometry Concepts

Additional Information: Applying Trigonometry

Trigonometry, especially the ratios and identities used here, is fundamental in various fields:

• Physics: Analyzing forces, waves, and motion.
• Engineering: Designing structures, calculating angles in mechanical systems.
• Navigation: Determining locations and distances using angles.
• Surveying: Measuring land, angles, and elevations.
• Computer Graphics: Rendering 3D scenes and animations.

Understanding how to manipulate trigonometric expressions and use identities is crucial for solving problems in these areas.