Miscellaneous Exercise 6: Review of Trigonometry

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Chapter Overview & SLOs

What is covered in the Miscellaneous Exercise? The Miscellaneous Exercise for Chapter 6 provides a comprehensive review of trigonometry, including trigonometric ratios, identities, heights and distances, bearings, and navigation problems.

How do we determine all trigonometric ratios from one given value? Use the fundamental identity $\sin^2 \theta + \cos^2 \theta = 1$ to find missing sine or cosine values.

• Step 1: Use $\sin^2 \theta + \cos^2 \theta = 1$ to find the missing ratio
• Step 2: Apply the ASTC rule to determine the correct sign based on quadrant
• Step 3: Find $\tan \theta = \sin \theta / \cos \theta$
• Step 4: Find reciprocals: $\csc \theta = 1/\sin \theta$, $\sec \theta = 1/\cos \theta$, $\cot \theta = 1/\tan \theta$

What is the ASTC rule for quadrant signs?

• Quadrant I (0°-90°): A = All ratios positive
• Quadrant II (90°-180°): S = Sine positive, others negative
• Quadrant III (180°-270°): T = Tangent positive, others negative
• Quadrant IV (270°-360°): C = Cosine positive, others negative

How do we simplify trigonometric identities? Use fundamental identities to simplify complex expressions.

• Example 1: $(1 - \cos^2 \theta) \sec^2 \theta = \sin^2 \theta \times \frac{1}{\cos^2 \theta} = \tan^2 \theta$
• Example 2: $\sec^2 \theta = 1 + \tan^2 \theta$
• Example 3: $\csc^2 \theta = 1 + \cot^2 \theta$

How do we solve real-world height and distance problems? Apply the tangent function $\tan \theta = \frac{h}{x}$.

• If angle of elevation $\theta$ and distance $x$ are known: $h = x \tan \theta$
• If height $h$ and angle $\theta$ are known: $x = h \cot \theta$
• Example: Building height $h = 50 \times \tan 30^\circ = 50 \times 0.5774 = 28.87$ m
• Hot air balloon: Rate of change = vertical displacement / time

How do we calculate three-figure bearings? Bearings are measured clockwise from North, always expressed as three digits.

• North = $000^\circ$, East = $090^\circ$, South = $180^\circ$, West = $270^\circ$
• Reciprocal bearing: Add or subtract $180^\circ$ (e.g., $250^\circ \rightarrow 250^\circ - 180^\circ = 070^\circ$)

How do we apply Pythagorean theorem to multi-triangle figures? For right triangles, $a^2 + b^2 = c^2$.

• Example: In a right triangle with legs $a = 6$ cm, $b = 8$ cm, hypotenuse $c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$ cm

How do we find geometric dimensions using area formulas? For a right triangle, $A = \frac{1}{2}ab$.

• If area and one leg are known, find the other leg
• Then use Pythagorean theorem to find the hypotenuse
• Example: In an isosceles right triangle, legs are equal → $A = \frac{1}{2}a^2$, so $a = \sqrt{2A}$

How do we analyze ratios in different quadrants? For Quadrant II ($90^\circ < \theta < 180^\circ$):

• $\sin \theta$ = positive
• $\cos \theta$ = negative
• $\tan \theta$ = negative

Key formulas to remember:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $1 + \tan^2 \theta = \sec^2 \theta$
• $1 + \cot^2 \theta = \csc^2 \theta$
• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

These solutions are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we simplify trigonometric identities? Apply basic relations like $\sec^2 \theta = 1 + \tan^2 \theta$, $\csc^2 \theta = 1 + \cot^2 \theta$, and $(1 - \cos^2 \theta) \sec^2 \theta = \tan^2 \theta$ to reduce complex algebraic-trigonometric expressions into simpler forms using fundamental identities.
• How do we analyze ratios in different quadrants? Determine the correct algebraic signs for $\sin \theta$, $\cos \theta$, and $\tan \theta$ by identifying the terminal ray's position using the ASTC rule (All Students Take Calculus), such as recognizing that in Quadrant II ($90^\circ < \theta < 180^\circ$), sine is positive while cosine and tangent are negative.
• How do we apply trigonometry to physics and navigation? Solve for speed and distance by modeling motion through angles of elevation (using $\tan \theta = h/x$) and three-figure bearings (clockwise from North, reciprocal bearings $\pm 180^\circ$), converting vertical displacement into rate of change for problems like hot air balloon speeds.
• How do we find geometric dimensions in right triangles? Use area formulas like $A = \frac{1}{2}ab$ combined with the Pythagorean theorem $a^2 + b^2 = c^2$ to find unknown side lengths and the hypotenuse, including in special cases like isosceles right triangles where $a = b$.

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