Trigonometry: Apotemi Single Topic Book

Trigonometry Overview

• Trigonometry is a core mathematical field used in engineering, physics, and astronomy.
• Effective understanding requires detailed explanations paired with numerous worked examples.

Origin

• The "Apotemi Single Topic Book" series dedicates this book specifically to Trigonometry.

Approach

• "Stepping Technique" is used to subdivide topics.
• Instruction is reinforced with "STEP-UP" tests.
• Material is bolstered with "STEP Strengthening" tests.
• A comprehensive review is provided at the end with "MARATHON TESTS".

Intended Audience

• The book is designed for eleventh-grade students, but it may also help those in the twelvth grade who are preparing for examinations.
• Users can follow the order of Trigonometry - 1, Trigonometry - 3, Trigonometry - 5 for eleventh grade curriculum compliance

Solutions

• Solutions can be found through the Apotemi telephone application or the online platform apotemivideo.frns.in.

Feedback

• Readers are encouraged to send comments and suggestions to [email protected]
• Contributions to improving forthcoming editions are welcome

Apotemi

• Contact Fatih Ihtiyaroglu directly at [email protected] or [email protected].

Directed Angles

• An angle is defined as the union of two rays that share a common starting point.
• An angle has direction based on the rotation of the initial ray to the terminal ray.

Degrees

• A degree is defined as the measure of a central angle subtended by an arc equal to 1/360th of the circumference of a circle.
• The symbol (°) indicates degrees.
• A minute is 1/60th of one degree (1').
• A second is 1/60th of one minute (1").
• Using division one can obtain a decimal of degrees, minutes and seconds.

Radians

• A radian is the measure of a central angle subtended by an arc equal to the radius of the circle.
• A full circle measures 2π radians.

Degree Conversion

• From degrees to radians : D/360= R/2π
• D/180= R/π can be used as shorthand
• Radian/degree transformations are essential for problem-solving

Principal Angle

• The principal angle is the smallest non-negative angle coterminal with the given angle.
• Use the formula α + 360°⋅k (k ∈ Z) to determine the principal angle if that angle has degree α.
• Use the formular α + 2kπ (k ∈ Z) if that angle has the radians as α.
• The principal angle is in the range of [0°, 360°) in degrees or [0, 2π) in radians.

Unit Circle

• The unit circle is a circle with a radius of 1 centered at the starting point (origin) on the Cartesian coordinate system.

Equation

• The equation for a unit circle = x^2 + y^2 = 1.
• The equation helps link geometric figures to trigonometric functions

Trigonometric Functions in a Right Triangle

• The trigonometric functions for a right triangle with A as one of the acute angles.
• sinA = Opposite/Hypotenuse
• cosA = Adjacent/Hypotenuse
• tanA = Opposite/Adjacent
• cotA = Adjacent/Opposite

Relationship

• The functions values conform to cos²α + sin²α = 1.
• tanα = sinα/cosα
• cotA = cosA/sinA
• Tangent and cotangent are complementary: tanα⋅cotα = 1.

Signs of Trigonometric Functions by Quadrant

• Signs of trigonometric functions vary depending on what quadrant, indicated by positive or negative signs

Quadrant 1

• Positive to both sine and cosine.
• All trigonometric functions will have positive values.

Quadrant 2

• Positive to sine, negative to cosine.
• Tangent and cotangent are negative

Quadrant 3

• Negative for both sine and cosine
• Tangent and cotangent functions will have positive values.

Quadrant 4

• Positive to cosine, negative to sine.
• Tangent and cotangent will have negative values.

Sine and Cosine

• Cosine is adjacent/hypotenuse, sine is opposite/hypotenuse on the coordinate plane
• Sine values are y axis
• Cosine values are x axis

Sine and Cosine Functions

• The cosine function (y=cosx) relates an angle to the x-coordinate of a point on the unit circle
• The range of both is between -1 and 1 (-1≤cosx≤1 and -1≤sinx≤1 for any real number)

Odd and Even functions

• tan = tan(-θ)
• cot = -cote
• Secant and cosecant functions are defined and related to cosine and sine.
• secx = 1/cosx
• cscx= 1/sinx
• Secant and cosecant values stem from cosine and sine

Conversion With 90°

• For angles greater than 90° , use the 180° or 360° to express them in terms of smaller angles.
• Trigonometric function identities stay the same
• The sign of the original trigonometric function in that quadrant determines the sign of the converted expression.

Application

• Use 270° and 90° to covert one trig function to another
• Must verify the sign of the original angle as a reference point

Example of Steps

• Apply reference marks for a circle with diameter to the y coordinates
• Circle shows the reference to a sign in each area

Angle Comparisons In Each Quadrant

• Trigonometric functions follow different ordering based on the quadrant on the cartesian coordinate system