trig lec 2

Significance of Trigonometry in Academic Admissions

• Trigonometry is a foundational topic for entry tests in Pakistan's top universities, including NUST, FAST, IBA (BBA and CS sides), NED, Comsats, GIKI, and PIEAS.

• A paper for these universities almost never exists without Trigonometry questions, and its importance cannot be understated.

• It is highly recommended that students leave 3 to 5 blank pages at the beginning of their notes specifically for a growing formula sheet and short-cuts to assist in quick review.

Sum and Difference of Angles

• Sine Formulas: In Sine formulas, the trigonometric functions alternate ($\sin$ and $\cos$), but the mathematical sign remains the same as the operator in the expression.

  • $\sin(\alpha + \beta) = \sin(\alpha)\cos(\beta) + \cos(\alpha)\sin(\beta)$

  • \sin(\alpha - \beta) = \sin(\alpha)\cos(\beta) - \n\cos(\alpha)\sin(\beta)

• Cosine Formulas: In Cosine formulas, the functions appear in pairs ($\cos\cos$ and $\sin\sin$), but the mathematical sign is always the opposite of the expression's operator.

  • $\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha)\sin(\beta)$

  • $\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)$

• Tangent Formulas:

  • $\tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)}$

  • \tan(\alpha - \beta) = \frac{\tan(\alpha) - \n\tan(\beta)}{1 + \tan(\alpha)\tan(\beta)}

Standard Angles and The Non-Calculator Table Trick

• In most admission tests (NUST, FAST, IBA, GIKI, Habib University, Comsats, UET), calculators are strictly prohibited. Calculators are generally only allowed in NED and PIEAS.

• To find values for standard angles ($0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$) quickly, use the following method:

  1. Write the numbers $0, 1, 2, 3, 4$ in a row.

  2. Divide each number by $4$.

  3. Take the square root of each result.

• Sine Row Results:

  • $\sin(0^\circ) = \sqrt{\frac{0}{4}} = 0$

  • $\sin(30^\circ) = \sqrt{\frac{1}{4}} = \frac{1}{2}$

  • $\sin(45^\circ) = \sqrt{\frac{2}{4}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

  • $\sin(60^\circ) = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$

  • $\sin(90^\circ) = \sqrt{\frac{4}{4}} = 1$

• Cosine and Tangent Rows:

  • For Cosine values, reverse the Sine row results ($90^\circ$ sine value becomes $0^\circ$ cosine value).

  • For Tangent, use $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$.

  • $\tan(90^\circ) = \frac{1}{0} = \infty$ (Infinity/Not Possible).

Inverse Trigonometric Functions

• Clarification on Reciprocals: Inverse functions ($\sin^{-1}(x)$) are not achieved by simply moving a function from the numerator to the denominator ($1/\sin(x)$). Reciprocating a term is mathematically distinct from the inverse function.

• Definition: An inverse trigonometric function is achieved when a function is moved to the other side of an equation without its angle.

• Process: In a normal function, you input an angle to get a numeric value. In an inverse function, you input a number to retrieve the angle.

  • Example: If $\sin(90^\circ) = 1$, then $\sin^{-1}(1) = 90^\circ$.

  • Example: $\sin^{-1}(\frac{\sqrt{3}}{2}) = 60^\circ$.

  • Example: $\tan^{-1}(1) = 45^\circ$.

Angle Breaking and Reference Angles

• For non-standard angles like $15^\circ$, break them into standard angles to apply sum/difference formulas.

  • Example: Calculating $\cos(15^circ)$:

  • Break into $\cos(45^\circ - 30^\circ)$.

  • Apply formula: $\cos(45^\circ)\cos(30^\circ) + \sin(45^\circ)\sin(30^\circ)$.

  • Substitute values: $(\frac{1}{\sqrt{2}})(\frac{\sqrt{3}}{2}) + (\frac{1}{\sqrt{2}})(\frac{1}{2}) = \frac{\sqrt{3}+1}{2\sqrt{2}}$.

  • Example: Calculating $\tan(15^\circ)$:

  • Break into $\tan(45^\circ - 30^\circ)$.

  • Result after rationalization: $2 - \sqrt{3}$.

• Rationalization Process: Since roots are generally not accepted in the denominator in advanced mathematics, multiply and divide by the conjugate of the denominator.

  • $\frac{\sqrt{3}-1}{\sqrt{3}+1} \times \frac{\sqrt{3}-1}{\sqrt{3}-1} = \frac{(\sqrt{3}-1)^2}{3-1} = \frac{3 - 2\sqrt{3} + 1}{2} = 2 - \sqrt{3}$.

Complementary Trigonometric Properties

• Sine and Cosine are complementary functions, meaning they share the same values for angles that sum to $90^\circ$.

• Property: $\sin(\theta) = \cos(90^\circ - \theta)$.

  • $\sin(30^\circ) = \cos(60^\circ) = \frac{1}{2}$

  • $\sin(20^\circ) = \cos(70^\circ)$

  • $\sin(19^\circ) = \cos(71^\circ)$

• Application Scenario: In expressions like $\sin(19^\circ)\cos(11^\circ) + \sin(71^\circ)\sin(11^\circ)$, identify that $\sin(71^\circ)$ is equivalent to $\cos(19^\circ)$ to form the $\sin(\alpha + \beta)$ identity.

Quadrants and ASTC Rules

• ASTC / All Silly Tom Cats / Add Sugar To Coffee:

  • First Quadrant ($0^\circ - 90^\circ$): ALL functions are positive.

  • Second Quadrant ($90^circ - 180^circ$): Sine and Cosecant are positive.

  • Third Quadrant ($180^circ - 270^circ$): Tangent and Cotangent are positive.

  • Fourth Quadrant ($270^circ - 360^circ$): Cosine and Secant are positive.

• Reference Angle Theory:

  • Horizontal Line ($180^\circ, 360^\circ$): The trigonometric function remains the same (e.g., $\sin \rightarrow \sin$).

  • Vertical Line ($90^\circ, 270^\circ$): The function changes to its co-function ($\sin \rightleftharpoons \cos$, $\tan \rightleftharpoons \cot$, $\sec \rightleftharpoons \csc$).

• Handling Negative Angles:

  • $\sin(-\theta) = -\sin(\theta)$

  • $\tan(-\theta) = -\tan(\theta)$

  • $\cos(-\theta) = \cos(\theta)$ (Cosine absorbs the negative sign).

Axis Behavior Short-cuts

• X-axis ($0^\circ, 180^\circ, 360^\circ$):

  • $\sin(\theta) = 0$

  • $\cos(\theta) = 1 \text{ or } -1$

  • $\tan(\theta) = 0$

• Y-axis ($90^\circ, 270^\circ$):

  • $\sin(\theta) = 1 \text{ or } -1$

  • $\cos(\theta) = 0$

  • $\tan(\theta) = \infty$

Fundamental Trigonometric Identities

• $\sin^2(\theta) + \cos^2(\theta) = 1$

• $1 + \tan^2(\theta) = \sec^2(\theta)$

• $1 + \cot^2(\theta) = \csc^2(\theta)$

• Mnemonic: Tangent and Secant share a relationship; Cotangent ($c$) and Cosecant ($c$) share a relationship.

Co-terminal Angles

• Definition: Angles that share the same initial and terminal sides but have different rotation counts or directions.

• Calculation:

  • If $\theta$ is positive: $\text{Co-terminal} = \theta + 360^\circ$

  • If $\theta$ is negative: $\text{Co-terminal} = \theta - 360^\circ$

• Example: $\text{Co-terminal of } 30^\circ = 390^\circ$ (one extra rotation) and $-330^circ$ (opposite direction).

Double, Half, and Triple Angle Formulas

• Double Angle Formulas:

  • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

  • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$

  • $\cos(2\theta) = 2\cos^2(\theta) - 1$

  • $\cos(2\theta) = 1 - 2\sin^2(\theta)$

  • $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

• Half Angle Formulas: Structure remains same as double angle, but angles are halved (e.g., $2\theta \rightarrow \theta$ and $\theta \rightarrow \theta/2$).

  • $\sin(\theta) = 2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})$

  • $1 + \cos(\theta) = 2\cos^2(\frac{\theta}{2})$

  • $1 - \cos(\theta) = 2\sin^2(\frac{\theta}{2})$

  • $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$

  • $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$

  • $\tan(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$

• Triple Angle Formulas:

  • $\sin(3\theta) = 3\sin(\theta) - 4\sin^3(\theta)$

  • $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$

  • $\tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)}$

Sum-to-Product and Product-to-Sum Formulas

• Sum-to-Product:

  • $\sin(u) + \sin(v) = 2\sin(\frac{u+v}{2})\cos(\frac{u-v}{2})$

  • $\sin(u) - \sin(v) = 2\cos(\frac{u+v}{2})\sin(\frac{u-v}{2})$

  • $\cos(u) + \cos(v) = 2\cos(\frac{u+v}{2})\cos(\frac{u-v}{2})$

  • $\cos(u) - \cos(v) = -2\sin(\frac{u+v}{2})\sin(\frac{u-v}{2})$

• Product-to-Sum: These are algebraic rearrangements of the formulas above using $\alpha$ and $\beta$.

Solving Ratios via Trigonometric Relations

• Mnemonic: Sum People Have Curly Brown Hair They Painted Black.

  • $\sin(\theta) = \frac{\text{Perpendicular}}{\text{Hypotenuse}}$

  • $\cos(\theta) = \frac{\text{Base}}{\text{Hypotenuse}}$

  • $\tan(\theta) = \frac{\text{Perpendicular}}{\text{Base}}$

• Pythagorean Theorem: $H^2 = P^2 + B^2$.

  • Validation: The Hypotenuse must always be the largest side in a right-angled triangle.

• Quadrant Awareness: When solving for a ratio (e.g., given $\sin(\theta) = -4/5$), determine the quadrant using given conditions to decide the correct sign ($\pm$) for other ratios like $\tan(\theta)$ or $\cos(\theta)$.

Large Angle Rule for $\frac{n\pi}{2}$

• To evaluate trig functions for very large angles in the form $\frac{n\pi}{2}$, divide the numeric constant ($n$) by $4$.

• Remainder Rules:

  • Remainder $1$: Equivalent to $90^\circ$

  • Remainder $2$: Equivalent to $180^\circ$

  • Remainder $3$: Equivalent to $270^\circ$

  • Remainder $4/0$: Equivalent to $360^\circ$

• Example: $\sin(409\frac{\pi}{2}) \rightarrow 409 \div 4 \text{ has a remainder of } 1$. Therefore, $\sin(409\pi/2) = \sin(90^\circ) = 1$.

Examination Strategy

• Time Per MCQ: Admission tests vary; NUST has $80$ Math MCQs, IBA has $50$ MCQs at $4$ marks each. If a problem takes more than $1$ minute to solve, skip it and use the "Next" button.

• Universal Principle for Identity Proofs: If presented with a complex identity to prove (e.g., $\frac{\sin(\theta)}{1-\cos(\theta)}$), instead of algebraic manipulation, substitute a standard angle like $30^\circ$ or $45^\circ$ into the question and then check the options for the same result.