1. Another way to prove an identity is to work on both sides of the given equation and arrive at equivalent expressions.
2. Angle and their Measurement
2.1. Trigonometry a branch of mathematics that focuses on the study of relationships between the sides and angles of triangles.
2.1.1. Angle
2.1.1.1. -An angle 𝛼 rotated in a counterclockwise direction is positive. -An angle 𝛽 rotated in a clockwise direction is negative.
2.1.1.2. 1. Acute angle – an angle whose measurement is between 0° and 90°. 2. Obtuse angle – an angle whose measurement is between 90° and 180°. 3. Right angle – an angle whose measurement is exactly 90°. 4. Straight angle – an angle whose measurement is exactly 180°. 5. Reflex angle – an angle whose measurement is between 180° and 360°. 6. Complementary Angles – two angles whose sum is 90°. 7. Supplementary Angles – two angles whose sum is 180°.
2.1.2. Measuring Angles
2.1.2.1. A radian measure (rad) is defined as the measure of a central angle that subtends an arc equal to 𝜋 of the circumference of a circle.
2.1.2.1.1. The radian and degree measures are related as follows; If r represents the radian measure and d represents degree measure of an angle, then, 𝑟 𝑑 __ = __ 𝜋 180°
2.1.3. LENGTH OF AN ARC
2.1.3.1. The length of any circular arc s is equal to the product of the radius r of the circle and the radian measure of the central angle 𝜃 it subtends.
2.2. The six Circular Function y. x _=sin. __=cos. r. r y. x _=tan. __=cot x. y r. r _=sec. __=csc y. y
2.2.1. The function of any angle 𝜃 is equal to plus (+) or minus (-) the same function of its reference angle 𝛼. That is, 𝑠𝑖𝑛 𝜃 = ± 𝑠𝑖𝑛 𝛼, 𝑐𝑜𝑠 𝜃 = ± 𝑐𝑜𝑠 𝛼,𝑡𝑎𝑛 𝜃 = ± 𝑡𝑎𝑛 𝛼 𝑐𝑠𝑐 𝜃 = ± 𝑐𝑠𝑐 𝛼, 𝑠𝑒𝑐 𝜃 = ± 𝑠𝑒𝑐 𝛼, 𝑐𝑜𝑡 𝜃 ± = 𝑐𝑜𝑡 𝛼
2.2.1.1. Solving radius using Pythagorean Theorem; r=√x²+y²
3. Trigonometric Functions
3.1. We can illustrate by the correspondence between 𝜃 and it’s terminal point 𝑃(𝜃) = 𝑃(𝑥, 𝑦) for an underlying circle of radius 1.
3.1.1. Application:
3.1.1.1. •Geometry •Physics •Engineering •Calculus •Mathemarics
4. Trigonometric Identities
4.1. IDENTITY if it is true for all values in its domain.
4.1.1. FUNDAMENTAL IDENTETIES • Reciprocal Identities • Quotient Identities • Phythagorean Identities • Co- function Identities • Even/Odd Identities • Sum and difference Identities • Double- angle Identities • Half- angle Identities
5. Inverse Trigonometric Function
5.1. Arcsine Function Arccosine Function Arctangent Function Arcsecant Function Arccosecant Function Arccotangent Function
5.1.1. GRAPHING TRIGONOMETRY FUNCTION
5.1.1.1. Amplitude
5.1.1.2. Period
5.1.1.3. Phase shift
5.1.1.4. Vertical shift
5.1.1.5. Domain & Range