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Basic Properties of Trigonometric Ratios With Examples

TL;DR

The basic properties of trigonometric ratios are the fixed relationships that tie the six ratios together — the reciprocal relations, the quotient relations, the Pythagorean identities, the sign each ratio takes in each quadrant, and the range of values each can hold. This article explains all five property groups, why ( \sin \theta ) never exceeds 1 while ( \tan \theta ) runs unbounded, and works through six examples.

What Are the Basic Properties of Trigonometric Ratios?

The basic properties of trigonometric ratios are the standing relationships among sine, cosine, tangent, cosecant, secant, and cotangent — the reciprocal links, the quotient links, the Pythagorean identities, the sign rules across the four quadrants, and the range of each ratio. They follow directly from the definitions of the six trigonometric ratios in a right triangle, and together they let you find any ratio once you know one.

Group the properties into five families. The rest of the article takes each in turn.

• Reciprocal relations — each ratio is the reciprocal of one other.
• Quotient relations — tangent and cotangent are quotients of sine and cosine.
• Pythagorean identities — three identities born from ( a^2 + b^2 = c^2 ).
• Sign in the quadrants — which ratios are positive where, captured by ASTC.
• Range of values — the bounds each ratio respects.

The Reciprocal and Quotient Relations

The first two property groups come straight from the definitions and are the most-used. A reciprocal is the result of dividing 1 by a quantity.

The reciprocal relations pair each primary ratio with one reciprocal ratio: [ \csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta} ]

Read the pairings carefully: cosecant pairs with sine (not cosine), and secant pairs with cosine. The "co-" prefix on cosecant is a naming quirk, not a clue to its partner. These same flips are collected in reciprocal identities.

The quotient relations express tangent and cotangent through sine and cosine: [ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta} ]

Together, the reciprocal and quotient relations mean that sine and cosine alone generate all six ratios. Know those two for an angle and the rest are arithmetic.

The Pythagorean Identities

The three Pythagorean identities are the heaviest-lifting properties in all of trigonometry, and each is the Pythagorean theorem wearing a trigonometric coat.

[ \sin^2 \theta + \cos^2 \theta = 1 ] [ 1 + \tan^2 \theta = \sec^2 \theta ] [ 1 + \cot^2 \theta = \csc^2 \theta ]

The first comes from taking a right triangle with hypotenuse ( c ), opposite ( a ), adjacent ( b ), so ( a^2 + b^2 = c^2 ), and dividing every term by ( c^2 ): [ \frac{a^2}{c^2} + \frac{b^2}{c^2} = 1 \implies \sin^2 \theta + \cos^2 \theta = 1 ]

How Are Trigonometric Ratios Signed in Each Quadrant?

A ratio's sign depends on where the angle's terminal side lands on the unit circle. A quadrant is one of the four regions the x- and y-axes cut the plane into, numbered anticlockwise from the top-right.

The pattern is captured by ASTC — "All, Sine, Tangent, Cosine" — naming the ratios that are positive in each quadrant:

What Is the Range of Each Trigonometric Ratio?

The range of a ratio is the set of values it can actually take.

• Sine and cosine are each a side divided by the hypotenuse, and no side can exceed the hypotenuse — so both stay within ([-1,1]). A value like ( \sin \theta = 1.4 ) is impossible.
• Cosecant and secant are the reciprocals of numbers in ([-1,1]), so they are never between −1 and 1: ( |\csc \theta| \geq 1 ) and ( |\sec \theta| \geq 1 ).
• Tangent and cotangent are unbounded — they run from −∞ to +∞.

Examples of Basic Properties of Trigonometric Ratios

Example 1

If ( \sin \theta = \frac{3}{5} ), find ( \csc \theta ).
[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{3/5} = \frac{5}{3} ]
Final answer: ( \csc \theta = \frac{5}{3} ).

Example 2

If ( \sin \theta = \frac{3}{5} ), find ( \cos \theta ).
Use the Pythagorean identity:
[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25} \implies \cos \theta = \frac{4}{5} ]
Final answer: ( \cos \theta = \frac{4}{5} ).

Example 3

Using the values from Example 2, find ( \tan \theta ).
[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{3/5}{4/5} = \frac{3}{4} ]
Final answer: ( \tan \theta = \frac{3}{4} ).

Example 4

Prove that ( (1+\tan^2 \theta)\cos^2 \theta = 1 ).
Replace ( 1+\tan^2 \theta ) with ( \sec^2 \theta ):
[ (1+\tan^2 \theta)\cos^2 \theta = \sec^2 \theta \cdot \cos^2 \theta = 1 ]
Final answer: the identity holds.

Example 5

Find ( \cos \theta ) if ( \sin \theta = \frac{12}{13} ) in Quadrant II.
Use the Pythagorean identity:
[ \cos^2 \theta = 1 - \left(\frac{12}{13}\right)^2 = \frac{25}{169} \implies \cos \theta = -\frac{5}{13} ]
Final answer: ( \cos \theta = -\frac{5}{13} ).

Example 6

Is ( \sec \theta = \frac{1}{2} ) possible?
( |\sec \theta| \geq 1 ), so this is impossible.
Final answer: no.

Key Takeaways

• The basic properties of trigonometric ratios are five groups: reciprocal, quotient, Pythagorean, sign-by-quadrant, and range.
• Sine and cosine generate all six ratios through the reciprocal and quotient relations.
• The three Pythagorean identities express essential relationships: ( \sin^2 \theta + \cos^2 \theta = 1 ) and its relatives.
• Sine and cosine live within ([-1,1]); secant and cosecant stay outside; tangent and cotangent are unbounded.