12 Questions
What is the sine function defined as?
The ratio of opposite side to hypotenuse in a right triangle
In trigonometry, angles are measured in:
Degrees
Which trigonometric function deals with the ratio of adjacent side to hypotenuse?
Cosine
Which formula states that sine of two angles adds to twice the sine of half the sum of the angles?
Sum formula for sine
Which trigonometric function is the reciprocal of sine?
Cosecant
How many radians does a circle contain?
$2\pi$
What is the period of a sine function?
2π radians
What is the amplitude of a sine function?
1
What is the formula for cosine of the sum of two angles?
$cos(a)cos(b) + sin(a)sin(b)$
What is the factor formula for cosine of the product of two angles?
$cos(a)cos(b) - sin(a)sin(b) = cos(a - b)$
What is the derivative of the sine function with respect to x?
$cos(x)/√2$
What are periodic functions?
Functions whose graphs repeat the same pattern at regular intervals
Study Notes
Sine Function
The sine function is one of the six basic trigonometric functions. It is defined as the ratio of the length of the side opposite the angle in a right triangle to the length of the hypotenuse. Mathematically, sine is represented by the symbol sin(x).
Units of Angles: Degree and Radian
Degrees
In trigonometry, angles are measured in degrees, which is a unitless measurement. A circle contains 360 degrees and covers 2π radians.
Radians
Another method of measuring angles is through radians. A radian is the measure of a central angle of a circle, which cuts off an arc equal in length to the radius of the circle.
Trigonometric Functions, Allied & Compound Angles
Trigonometric functions are mathematical operations used to find the values of certain unknown sides or angles of triangles. Besides sine, cosine, and tangent, there are four more functions which are foundations for complex numbers, calculus, and other areas of mathematics. They are cosecant, secant, cotangent, and arccosine.
Allied and compound angles deal with the combination of two or more angles. The sum formulae for sine and cosine state that the sine and cosine of the sum of two angles add to twice the sine of half the sum and twice the cosine of half the difference, respectively. Product formulas involve subtraction, dividing by 2, adding the signs, and using the identities for the sine and cosine of angles and half angles.
Periodic Trigonometric Function
Periodic functions are functions whose graphs repeat the same pattern at regular intervals. Sine and cosine are periodic functions. The period of a sine function is 2π radians or 360°, and the amplitude of a sine function is 1.
Sum and Factor Formulae
Sum Formulae
Sum formulas are used to find the cosines of the sum or difference of two angles. For sum and difference formulae, the formula is:
cos(a ± b) = cos(a)cos(b) ± sin(a)sin(b)
For the derivative of the sine function, the formula simplifies to sin'(x) = cos(x)*θ(x) = cos(x)/√(1 − x^2).
Factor Formulae
Factor formulae are used to find the cosines of product and quotient of two angles. The factor formula for cosine is:
cos(a)cos(b) ± sin(a)sin(b) = cos(a ± b) or sin(a)sin(b) ± cos(a ± b) = 0
For the derivative of the sine function, the formula simplifies to:
sin'(x) = cos(x)/√2
These formulas can help solve problems that involve trigonometric functions, such as finding missing sides or angles of triangles.
Test your knowledge on trigonometric functions such as sine, cosine, tangent, and more, along with allied and compound angles. Explore unit conversions between degrees and radians, learn about sum and factor formulae, and understand periodic functions in trigonometry.
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