Trigonometry Basics: Angles and Ratios

Trigonometry Basics

Angles and Measurements

• An angle is formed by two rays sharing a common endpoint (vertex)
• Angles can be measured in degrees (°), radians, or gradients
• 1 radian is the angle subtended by an arc of length 1 unit in a unit circle

Trigonometric Ratios

Sine (sin), Cosine (cos), and Tangent (tan)

• sin(A) = opposite side / hypotenuse
• cos(A) = adjacent side / hypotenuse
• tan(A) = opposite side / adjacent side

Reciprocal Identities

• cosec(A) = 1/sin(A)
• sec(A) = 1/cos(A)
• cot(A) = 1/tan(A)

Pythagorean Identity

• sin^2(A) + cos^2(A) = 1

Trigonometric Identities

Sum and Difference Formulas

• sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

• cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

• tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))

• sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

• cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

• tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Double and Half Angle Formulas

• sin(2A) = 2sin(A)cos(A)

• cos(2A) = cos^2(A) - sin^2(A)

• tan(2A) = 2tan(A) / (1 - tan^2(A))

• sin(A/2) = ±√((1 - cos(A))/2)

• cos(A/2) = ±√((1 + cos(A))/2)

• tan(A/2) = ±√((1 - cos(A))/(1 + cos(A)))

Applications of Trigonometry

Right Triangles

• Solving for missing sides and angles in right triangles

Oblique Triangles

• Solving for missing sides and angles in oblique triangles using law of sines and law of cosines

Graphs of Trigonometric Functions

• sine, cosine, and tangent functions have periodic graphs with amplitude and phase shifts

Angles and Measurements

• An angle is formed by two rays sharing a common endpoint called the vertex
• Angles can be measured in three different units: degrees, radians, and gradients
• One radian is equivalent to the angle subtended by an arc of length 1 unit in a unit circle

Trigonometric Ratios

• The sine of an angle is the ratio of the opposite side to the hypotenuse
• The cosine of an angle is the ratio of the adjacent side to the hypotenuse
• The tangent of an angle is the ratio of the opposite side to the adjacent side
• These ratios can be used to find the length of sides and the measure of angles in right triangles

Reciprocal Identities

• The cosecant of an angle is the reciprocal of the sine
• The secant of an angle is the reciprocal of the cosine
• The cotangent of an angle is the reciprocal of the tangent

Pythagorean Identity

• The sum of the squares of the sine and cosine of an angle is equal to 1
• This identity can be used to find the value of one trigonometric function if the value of the other is known

Trigonometric Identities

• The sum and difference formulas for sine, cosine, and tangent can be used to find the value of these functions for the sum or difference of two angles
• The double and half angle formulas for sine, cosine, and tangent can be used to find the value of these functions for twice or half an angle

Sum and Difference Formulas

• The sine of the sum of two angles is the product of the sines and cosines of the individual angles
• The cosine of the sum of two angles is the product of the cosines minus the product of the sines
• The tangent of the sum of two angles is the sum of the tangents divided by 1 minus the product of the tangents

Double and Half Angle Formulas

• The sine of twice an angle is twice the product of the sine and cosine of the angle
• The cosine of twice an angle is the square of the cosine minus the square of the sine
• The tangent of twice an angle is twice the tangent divided by 1 minus the square of the tangent
• The sine of half an angle is plus or minus the square root of half of the difference between 1 and the cosine of the angle
• The cosine of half an angle is plus or minus the square root of half of the sum of 1 and the cosine of the angle
• The tangent of half an angle is plus or minus the square root of the difference between 1 and the cosine of the angle divided by the sum of 1 and the cosine of the angle

Applications of Trigonometry

• Trigonometry can be used to solve problems involving right triangles, including finding the length of missing sides and the measure of missing angles
• Trigonometry can be used to solve problems involving oblique triangles, including finding the length of missing sides and the measure of missing angles using the law of sines and law of cosines
• The graphs of trigonometric functions, such as sine, cosine, and tangent, have periodic properties, including amplitude and phase shifts, which can be used to model real-world phenomena.

Learn the fundamentals of trigonometry, including angles, measurements, and trigonometric ratios such as sine, cosine, and tangent.