5.1.1 A Review of Trigonometry

Question 1

A Review of Trigonometry

Answer 1

• The Pythagorean theorem relates the sides of a right triangle. Each trigonometric function matches the ratio of two sides of a right triangle to one of the angles.
• The Pythagorean identity is sin^2x+cos^2x=1

Question 2

note

Answer 2

• Trigonometry examines the relationships between the lengths of the sides of a right triangle and one of the acute angles of that triangle.
• The most important identity to remember in trigonometry is the Pythagorean theorem. This theorem is the foundation of trigonometry.
• The three most basic trig functions are the sine function, the cosine function, and the tangent function. Each is defined for a given angle by the ratio of two of the sides of a right triangle.
• Notice that the tangent function can be expressed in terms of the sine and cosine functions.
• The reciprocal trig functions are defined by reciprocal
  relationships with the three basic trig functions. The
  reciprocal functions are the cosecant function, the
  secant function, and the cotangent function.
• Notice that the reciprocal of sin θ is csc θ, not sec θ.
• There are some notation conventions that you should be aware of regarding trig functions. Notice that if a trig function is raised to a power, then the exponent can either appear after the name of the trig function or outside of the function surrounded, by parentheses. If you do not include the parentheses, it is easy to confuse the meaning of the exponent.
• There are many relationships between the different
  trigonometric functions. The most fundamental is the
  Pythagorean identity.
• The Pythagorean identity states that the square of the sine of any angle plus the square of the cosine of that angle is equal to 1.
• You can prove the Pythagorean identity using the Pythagorean theorem.

Question 3

Simplify the expression: sin(x)/cos(x) ⋅ cot(x)

Answer 3

1

Question 4

Simplify the expression: sin(x) ⋅ sec(x) ⋅ tan(x) − sec^2(x)

Answer 4

-1

Question 5

Simplify the expression: sec(x) ⋅ csc(x) ⋅ cot(x)

Answer 5

csc^2 (x)

Question 6

Simplify the expression: cscx/sinx−cot^2x

Answer 6

1

Question 7

Simplify the expression:sinx⋅1/cosx

Answer 7

tan x

Question 8

Simplify the expression:cosx⋅1/sinx

Answer 8

cot x

Question 9

Simplify the expression:sin(x)⋅1/csc(x)+cos(x)⋅1/sec(x)

Answer 9

1

Question 10

Simplify the expression:tanx⋅cosx

Answer 10

sin x

Question 11

Simplify the expression:sin(x)⋅1/sec(x)−cos(x)⋅1/csc(x)

Answer 11

0

Question 12

Evaluate the following expression exactly using trigonometric identities.sin^2 π/10+cos^2 π/10

Answer 12

1

Question 13

Simplify the expression:tanx⋅cos^2x

Answer 13

sin x cos x