This trigonometry lesson defines the basic functions, states what the most common identities are, and then proves them. The rest of the lesson shows some basic tips and tricks that may be used to manipulate trigonometric functions: these are useful when the math student is asked to prove a trigonometric identity (of the form Left Hand Side = Right Hand Side), or when a trigonometric function needs to be manipulated into a form that is easier to use.
Basic Trigonometric Definitions
The image in this article shows a right-angled triangle. The basic parts are the hypotenuse (H) - the longest of the triangle sides, the included angle theta (θ), the side opposite to θ (O), and the side adjacent to θ (A).
By definition,
sineθ = O / H (Usually abbreviated to sinθ)
cosineθ = A / H (Usually abbreviated to cosineθ)
tangentθ = O / A (Usually abbreviated to tanθ)
Some other trigonometric functions are
cosecantθ = 1 / sinθ (Usually abbreviated to cosecθ)
secantθ = 1 / cosθ (Usually abbreviated to secθ)
cotanθ = 1 / tanθ (Usually abbreviated to cotθ)
Basic Trigonometric Function Identities
It is known from Pythagoras Theorem that
O² + A² = H²
Dividing through by H² gives
(O/H)² + (A/H)² = 1, but since O/H = sinθ and A/H = cosθ then
sin²θ + cos²θ = 1
Dividing this throughout now by cos²θ gives
tan²θ + 1 = sec²θ
Dividing throughout by sin²θ gives
cot²θ + 1 = cosec²θ
These are the main identities used in trigonometry.
There are a few others that are used, called the "Odd-Even" identities:
sin(-θ) = -sinθ
cos(-θ) = cosθ
tan(-θ) = -tanθ
The last set of trigonometric identities is the set of "Double-angle" identities:
sin2θ = 2sinθcosθ (See also De Moivre's Theorem)
cos2θ = cos²θ - sin²θ (and since sin²θ + cos²θ = 1, this may also be stated as 1 - 2sin²θ, or 2cos²θ - 1)
tan2θ = 2tanθ / (1 - tan²θ)
Trigonometry Lesson - Tips To Solve and Simplify Expressions
Here are some tips that may be used to manipulate or simplify trigonometric expressions. Each is stated simply, with a brief explanation.
- Questions that require a trigonometric identity to be proven are usually of the form Left Hand Side = Right Hand Side. The objective is not to find a value of θ for which it is true. The objective is to prove that the equation is true for ALL values of θ.
- Leave one side of the statement alone, and manipulate the other side to prove that it is equivalent. Very often, it is easier to do the proof by leaving the most simple expression alone, and simplify the other.
- It is useful to remember the formula for the difference of two squares: (a² - b²) = (a + b)(a - b). This is used in many trigonometric identity proofs.
- It is often the tendancy of students to simplify expressions. Sometimes the opposite needs to be done e.g. Convert "1" into sin²θ + cos²θ
- Recognize key "pairs" like 1 + sin2θ. Expanding both of these into (sin²θ + cos²θ + 2sinθcosθ) allows the term to become (sinθ + cosθ)².
- Try to eliminate denominators, where possible. For example, the term 1 / (1 + sinθ) may be simplified by multiplying numerator and denominator by 1 - sinθ. The denominator then becomes 1 - sin²θ, or cos²θ, which may then be divided through the numerator.
- This tip is by far the most important: Practice, practice, and practice. It is only with copious amounts of practice that trigonometric proofs and algebraic manipulation become fluent and easy.
Trigonometric Identities and Proofs Summary
The main trigonometric definitions and identities are shown here, along with several tips and hints about how to use them to good effect. Algebraic simplification and manipulation are key to successful math study, and this becomes more true as the higher levels of mathematics are attempted. The article "De Moivres Theorem Examples Cos 3x, Sin3x, Cos4x and Sin4x" shows examples of where trigonometric function manipulation is useful. The formulas for the sin(a+b), cos(a+b), sin(a)+sin(b) and cos(a)+cos(b) are derived in another article at this site.
Trigonometric Identities and Proofs References
Most of the identities listed here have been derived from Pythagoras Theorem. The author is an experienced math teacher at all levels up to and including college level. The tips and hints listed here were learned coaching students over many years. A worked example of a trigonometric identity that could not be solved in the forum www.physicsforum.com is described in "Trigonometric Identity Advanced Example".
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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