Comprehensive Trigonometry 1 Review and Problem Solving Guide

Fundamental Ranges and Extrema of Trigonometric Functions

The fundamental trigonometric functions, sine and cosine, are bounded by the interval of negative one to one. Specifically, for any real number $x$, the values are constrained such that $-1 \le \sin(x) \le 1$ and $-1 \le \cos(x) \le 1$. These limits are defined by the projection of points on the unit circle onto the vertical and horizontal axes, respectively.

When dealing with linear combinations of sine and cosine in the form $a \cdot \sin(x) + b \cdot \cos(x)$, where $a$ and $b$ are real numbers, the function oscillates between a specific range. Detailed analysis shows that the maximum value of this expression is given by $\sqrt{a^2+b^2}$, while the minimum value is $-\sqrt{a^2+b^2}$. This result is derived from the vector dot product or the harmonic addition theorem.

Unlike sine and cosine, the tangent and cotangent functions are derived as ratios that involve potential division by zero at specific points, corresponding to the vertical and horizontal asymptotes on the unit circle. Consequently, these functions have no upper or lower bounds. The range for both tangent and cotangent is the set of all real numbers, expressed as -\infty < \tan(\alpha) < \infty and -\infty < \cot(\alpha) < \infty.

Basic Trigonometric Identities and Relationships

Several fundamental identities serve as the building blocks for simplifying complex trigonometric expressions. The Pythagorean identity states that for any angle $x$, the sum of the squares of the sine and cosine functions is always equal to unity: $\sin^2(x) + \cos^2(x) = 1$. Additionally, the relationship between tangent and cotangent is reciprocal, meaning their product is always one: $\tan(x) \cdot \cot(x) = 1$.

Quadrants and Sign Definitions

The coordinate plane is divided into four quadrants, each defining the sign (positive or negative) of the trigonometric functions based on the position of the terminal side of the angle $x$. These are categorized as follows:

In the first quadrant (0^\circ < x < 90^\circ), all primary trigonometric functions are positive: $\sin(x) = +$, $\cos(x) = +$, $\tan(x) = +$, and $\cot(x) = +$.

In the second quadrant (90^\circ < x < 180^\circ), only the sine function and its reciprocal are positive: $\sin(x) = +$, while $\cos(x) = -$, $\tan(x) = -$, and $\cot(x) = -$.

In the third quadrant (180^\circ < x < 270^\circ), the tangent and cotangent functions are positive due to the ratio of two negative coordinates: $\sin(x) = -$ and $\cos(x) = -$, while $\tan(x) = +$ and $\cot(x) = +$.

In the fourth quadrant (270^\circ < x < 360^\circ), only the cosine function and its reciprocal are positive: $\cos(x) = +$, while $\sin(x) = -$, $\tan(x) = -$, and $\cot(x) = -$.

Angle Reduction and Transformation Formulas

Reduction formulas allow for the expression of trigonometric functions of any angle in terms of an acute angle $\alpha$. When the transformation involves the horizontal axis ($\pi$ or $2\pi$), the function name remains the same, but the sign is adjusted based on the quadrant. Examples include $\sin(\pi - \alpha) = \sin(\alpha)$, $\cos(\pi - \alpha) = -\cos(\alpha)$, $\tan(\pi - \alpha) = -\tan(\alpha)$, and $\cot(\pi - \alpha) = -\cot(\alpha)$. Similarly, for the third quadrant, $\sin(\pi + \alpha) = -\sin(\alpha)$ and $\cos(\pi + \alpha) = -\cos(\alpha)$, while $\tan(\pi + \alpha) = \tan(\alpha)$.

When transformations involve the vertical axis ($\frac{\pi}{2}$ or $\frac{3\pi}{2}$), the function changes to its co-function (sine becomes cosine, tangent becomes cotangent, and vice-versa) in addition to the sign adjustment. For instance, $\sin(\frac{\pi}{2} - \alpha) = \cos(\alpha)$, $\cos(\frac{\pi}{2} - \alpha) = \sin(\alpha)$, and $\tan(\frac{\pi}{2} - \alpha) = \cot(\alpha)$. Advanced variations include $\sin(\frac{3\pi}{2} + \alpha) = -\cos(\alpha)$, $\cos(\frac{3\pi}{2} + \alpha) = \sin(\alpha)$, and $\cot(\frac{3\pi}{2} + \alpha) = -\tan(\alpha)$.

Algebraic Problem Solving in Trigonometry

Practical applications often require manipulating algebraic expressions. One scenario involves finding the value of $\tan^2(x) + \cot^2(x)$ given that $\tan(x) - \cot(x) = 3$. By squaring both sides of the given equation, $\tan^2(x) - 2\tan(x)\cot(x) + \cot^2(x) = 9$. Using the identity $\tan(x)\cot(x) = 1$, we obtain $\tan^2(x) - 2 + \cot^2(x) = 9$, which leads to the result $\tan^2(x) + \cot^2(x) = 11$.

Another example involves the simplification of rational expressions. Consider the expression $\frac{\cos(x) \cdot \sin(3x) + \sin(x) \cdot \cos(3x)}{1 - \sin^2(x)}$. The numerator is an expansion of the sine addition formula $\sin(A+B)$, specifically $\sin(3x+x) = \sin(4x)$. The denominator simplifies to $\cos^2(x)$ via the Pythagorean identity. Therefore, the expression simplifies into its most reduced form based on these substitutions.

In cases where $\sin(x) - \cos(x) = \frac{1}{2}$, we can determine the value of $\sin^4(x) + \cos^4(x)$. Squaring the initial equation yields $\sin^2(x) - 2\sin(x)\cos(x) + \cos^2(x) = \frac{1}{4}$. Substituting the identity $\sin^2(x) + \cos^2(x) = 1$ gives $1 - 2\sin(x)\cos(x) = \frac{1}{4}$, which means $2\sin(x)\cos(x) = \frac{3}{4}$ and $\sin(x)\cos(x) = \frac{3}{8}$. The expression $\sin^4(x) + \cos^4(x)$ is equal to $(\sin^2(x) + \cos^2(x))^2 - 2\sin^2(x)\cos^2(x) = 1^2 - 2(\frac{3}{8})^2 = 1 - 2(\frac{9}{64}) = 1 - \frac{9}{32} = \frac{23}{32}$.

Geometric Applications and the Unit Circle

Trigonometry is deeply integrated with geometry, particularly within the unit circle. For a point $K$ on the unit circle defined by an angle $\alpha$, its coordinates can be represented in various forms depending on the reference angle. If an angle is given as $m(BOL) = \alpha$ in the context of specific rotational shifts, the coordinates of a point $K$ might be expressed as $(\cos(270^\circ + \alpha), \sin(270^\circ + \alpha))$, which utilizes the full circular rotation properties.

Length calculations in the unit circle often involve identifying ratios. In a configuration where $m(EOC) = x$, the ratio of segment lengths $\frac{|DE|}{|AB|}$ can be evaluated. Using the definitions of trigonometric segments (where tangent and secant are lines exterior to the circle), such ratios frequently simplify to functions like $\cos(x)$ or $\sin(x)$.

In right-angled triangle problems (ABC), given an altitude $[AD]$ perpendicular to the hypotenuse $[BC]$ and a side length such as $|AC| = 1$, the length of other segments like $|BD|$ can be derived. By analyzing the similar triangles formed and the angle $x$, it is found that $|BD| = \frac{\sin^2(x)}{\cos(x)}$ or other related variations like $\tan(x) \sin(x)$.

Analysis of Trigonometric Expressions in Specific Domains

Analyzing the possibility of expressions equaling zero requires checking the signs within a specific interval. For the interval 180^\circ < x < 270^\circ (the third quadrant):

Expression I: $a + b = \sin(x) + \cos(x)$. Since both sine and cosine are negative in the third quadrant, their sum must be negative and can never be zero.

Expression II: $a \cdot b - c = \sin(x)\cos(x) - \tan(x)$. In this quadrant, $\sin(x)\cos(x)$ is positive (negative times negative) and $\tan(x)$ is also positive. Thus, their difference could potentially be zero depending on the specific value of $x$.

Expression III: $a^2 + c = \sin^2(x) + \tan(x)$. Since the square of sine is positive and tangent is positive in the third quadrant, their sum is strictly positive and cannot be zero.

Ordering and Signs of Trigonometric Values

Comparing trigonometric values requires reducing all angles to the first quadrant. For example, to order $a = \tan(221^\circ)$, $b = \sin(139^\circ)$, $c = \cos(139^\circ)$, and $d = \cot(139^\circ)$: $a = \tan(180^\circ + 41^\circ) = \tan(41^\circ)$ (positive). $b = \sin(180^\circ - 41^\circ) = \sin(41^\circ)$ (positive). $c = \cos(180^\circ - 41^\circ) = -\cos(41^\circ)$ (negative). $d = \cot(180^\circ - 41^\circ) = -\cot(41^\circ)$ (negative). Since \tan(41^\circ) > \sin(41^\circ) for acute angles, a > b. For the negative values, comparisons depend on the growth of the absolute values of cosine and cotangent.

In signaling problems, such as determining the signs of $x = \tan(140^\circ) - \tan(200^\circ)$, $y = \sin(40^\circ) \cdot \cos(40^\circ)$, and $z = \tan(70^\circ) - \cot(60^\circ)$, one must evaluate each term. Since $140^\circ$ is Q2 (tan is negative) and $200^\circ$ is Q3 (tan is positive), $x = (-) - (+)$ which is negative. Since $40^\circ$ is Q1, $y = (+) \cdot (+)$ which is positive.

Advanced Trigonometric Equations and Conditions

When given conditions like $x + 2y = \frac{\pi}{2}$, complex ratios like $\frac{\sin(x+y)}{\cos(y)} - \frac{\tan(x-y)}{\cot(3y)}$ can be evaluated. From the condition, $x+y = \frac{\pi}{2} - y$, meaning $\sin(x+y) = \sin(\frac{\pi}{2} - y) = \cos(y)$. Thus, the first term $\frac{\cos(y)}{\cos(y)}$ is $1$. Similarly, the second term can be analyzed by substituting $\frac{\pi}{2} - 2y$ for $x$, leading to a total value for the expression, often resulting in $0$ or $2$ depending on the specific variables.

Equations involving squared terms and cross-products, such as $2 \sin^2(x) + \sin(x)\cos(x) - \cos^2(x) = 0$, can be solved by dividing by $\cos^2(x)$ to form a quadratic in terms of $\tan(x)$: $2 \tan^2(x) + \tan(x) - 1 = 0$. Factoring this leads to $\tan(x) = \frac{1}{2}$ or $\tan(x) = -1$, allowing for the determination of $\sin(x) \cdot \cos(x)$ by constructing representative triangles.