Trigonometric Functions: Definitions, Identities, and Signs

Trigonometric Functions

Introduction to Trigonometric Functions

Trigonometric functions are fundamental mathematical tools used to relate the angles of a triangle to the lengths of its sides. They are widely applied in geometry, physics, engineering, and many other fields. The six primary trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Using the Definitions of the Trigonometric Functions

Reciprocal Identities

Reciprocal identities express each trigonometric function in terms of its reciprocal. These identities are valid for all angles $\theta$ for which both functions are defined.

• Sine and Cosecant: $\sin\theta = \dfrac{1}{\csc\theta}$, $\csc\theta = \dfrac{1}{\sin\theta}$

• Cosine and Secant: $\cos\theta = \dfrac{1}{\sec\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$

• Tangent and Cotangent: $\tan\theta = \dfrac{1}{\cot\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$

Example: Using Reciprocal Identities

• Given: $\sec\theta = \dfrac{5}{3}$

• Find: $\cos\theta$

  • $\cos\theta = \dfrac{1}{\sec\theta} = \dfrac{1}{5/3} = \dfrac{3}{5}$

• Given: $\csc\theta = -\dfrac{\sqrt{12}}{2}$

• Find: $\sin\theta$

  • $\sin\theta = \dfrac{1}{\csc\theta} = \dfrac{1}{-\sqrt{12}/2} = -\dfrac{2}{\sqrt{12}}$

  • Rationalize denominator: $\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}$

  • $-\dfrac{2}{2\sqrt{3}} = -\dfrac{1}{\sqrt{3}} = -\dfrac{\sqrt{3}}{3}$

Signs and Ranges of Function Values

Signs of Trigonometric Functions in Each Quadrant

The sign of each trigonometric function depends on the quadrant in which the angle's terminal side lies. The following table summarizes the signs:

Quadrant	sin $\theta$	cos $\theta$	tan $\theta$	cot $\theta$	sec $\theta$
I	+	+	+	+	+
II	+	-	-	-	-
III	-	-	+	+	-
IV	-	+	-	-	+

Quadrant Diagram

• Quadrant I: All functions positive ($x > 0, y > 0$)

• Quadrant II: Sine and cosecant positive ($x < 0, y > 0$)

• Quadrant III: Tangent and cotangent positive ($x < 0, y < 0$)

• Quadrant IV: Cosine and secant positive ($x > 0, y < 0$)

Examples: Determining Signs of Function Values

• Angle: 77°

  • 77° lies in Quadrant I. All trigonometric function values are positive.

• Angle: 300°

  • 300° lies in Quadrant IV. Cosine and secant are positive; sine, cosecant, tangent, and cotangent are negative.

• Angle: 200°

  • 200° lies in Quadrant II. Sine and cosecant are positive; all other function values are negative.

Pythagorean and Quotient Identities

Pythagorean Identities

Pythagorean identities relate the squares of sine, cosine, tangent, and their reciprocals. They are derived from the Pythagorean Theorem.

• $\sin^2\theta + \cos^2\theta = 1$

• $\tan^2\theta + 1 = \sec^2\theta$

• $1 + \cot^2\theta = \csc^2\theta$

Quotient Identities

Quotient identities express tangent and cotangent in terms of sine and cosine.

• $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

• $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$

Ranges of Trigonometric Functions

Range of Each Function

• $\sin\theta, \cos\theta$: $[-1, 1]$

• $\tan\theta, \cot\theta$: All real numbers $(-\infty, \infty)$

• $\sec\theta, \csc\theta$: $(-\infty, -1] \cup [1, \infty)$

Example: Deciding Whether a Value Is in the Range

• Is $\sin\theta = 2.5$ possible?

  • No, because $\sin\theta$ must be between -1 and 1.

• Is $\tan\theta = 110.47$ possible?

  • Yes, tangent can take any real value.

• Is $\sec\theta = 0.6$ possible?

  • No, because $|\sec\theta| \geq 1$ for all defined $\theta$.

Finding All Function Values Given One Value and the Quadrant

Procedure

Given one trigonometric function value and the quadrant, you can find the remaining function values using identities and the signs determined by the quadrant.

• Use the Pythagorean identity to find the missing value.

• Assign the correct sign based on the quadrant.

• Use reciprocal and quotient identities to find other functions.

Example: Given $\sin\theta = \dfrac{2}{3}$ in Quadrant II

• Let $y = 2$, $r = 3$. Find $x$ using $x^2 + y^2 = r^2$:

• $x^2 + 2^2 = 3^2 \Rightarrow x^2 = 9 - 4 = 5 \Rightarrow x = -\sqrt{5}$ (negative in Quadrant II)

• $\cos\theta = \dfrac{x}{r} = \dfrac{-\sqrt{5}}{3}$

• $\tan\theta = \dfrac{y}{x} = \dfrac{2}{-\sqrt{5}} = -\dfrac{2\sqrt{5}}{5}$

• $\csc\theta = \dfrac{r}{y} = \dfrac{3}{2}$

• $\sec\theta = \dfrac{r}{x} = -\dfrac{3}{\sqrt{5}} = -\dfrac{3\sqrt{5}}{5}$

• $\cot\theta = \dfrac{x}{y} = -\dfrac{\sqrt{5}}{2}$

Summary Table: Signs of Trigonometric Functions by Quadrant

Quadrant	Positive Functions	Negative Functions
I	All	None
II	sin, csc	cos, sec, tan, cot
III	tan, cot	sin, csc, cos, sec
IV	cos, sec	sin, csc, tan, cot

Additional info: These notes expand on the provided slides and images, adding full definitions, examples, and context for each identity and property. The examples and tables are reconstructed and clarified for academic completeness.