Basic Trigonometry

Trigonometry connects algebra and geometry. While it deals with triangle ratios, its algebraic identities are essential for simplifying expressions and solving equations.

The Right Triangle Definitions

For an acute angle $θ$ in a right triangle:

$sin θ = \frac{Opposite}{Hypotenuse}$
$cos θ = \frac{Adjacent}{Hypotenuse}$
$tan θ = \frac{Opposite}{Adjacent} = \frac{s i n θ}{c o s θ}$

Reciprocal Functions

$csc θ = \frac{1}{s i n θ}$
$sec θ = \frac{1}{c o s θ}$
$cot θ = \frac{1}{t a n θ}$

Fundamental Identities

Theorem (Pythagorean Identities)

sin^{2} θ + cos^{2} θ = 1

$1 + tan^{2} θ = sec^{2} θ$
$1 + cot^{2} θ = csc^{2} θ$

Proof (Visual Proof). In a right triangle with hypotenuse 1, the legs are $sin θ$ and $cos θ$ . By Pythagoras, leg $^{2}$ + leg $^{2}$ = hyp $^{2}$ , so $sin^{2} θ + cos^{2} θ = 1$ . ∎

Cofunction Identities

"Co" stands for "complementary."

$sin (9 0^{\circ} - θ) = cos θ$
$tan (9 0^{\circ} - θ) = cot θ$
$sec (9 0^{\circ} - θ) = csc θ$

Olympiad Applications: Algebraic Manipulation

Trig substitutions can simplify algebraic expressions, even when no angles are mentioned.

Tip (Trig Substitution)

* If you see

1 - x^{2}

, try

x = sin θ

x = cos θ

If you see $1 + x^{2}$ , try $x = tan θ$ .
If you see $x^{2} - 1$ , try $x = sec θ$ .

Practice Problems

Exercise (Problem 1)

Simplify the expression:

\frac{s i n ^{3} θ + c o s ^{3} θ}{s i n θ + c o s θ} + sin θ cos θ

Exercise (Problem 2)

tan θ + cot θ = 4

, find the value of

sin θ cos θ

Exercise (Problem 3)

Solve for

x

in terms of

θ

x = a^{2} cos^{2} θ + b^{2} sin^{2} θ

and

a = b