Class 10 Mathematics

Trigonometry is the branch of mathematics that deals with the study of relationships between sides and angles of a triangle. It is derived from Greek words, Tri meaning Three, Gon means Angle and Metron means Measure.

The ratio of lengths of two sides of a right-angled triangle is called a Trigonometric Ratio.

3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions and today are pervasive in both pure and applied mathematics. Trigonometry is also the foundation of surveying.

Trigonometric ratios

There are six trigonometric ratios.

Consider a right angled triangle ABC with :

• AC = hypotenuse of triangle.
• AB = side adjacent to angle A.
• BC = side opposite to angle A.

Trigonometric ratios of right angled triangle ABC with span id="MathJax-Element-1-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="B=90">∠B=90∘∠B=90∘:

• sin A = sine of span id="MathJax-Element-2-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="A">∠A∠A = span id="MathJax-Element-3-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="(sideoppositetoA)hypotenuse">(sideoppositeto∠A)hypotenuse(sideoppositeto∠A)hypotenuse= span id="MathJax-Element-4-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="BCAC">BCACBCAC
• cos A = cosine of span id="MathJax-Element-5-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="A">∠A∠A = span id="MathJax-Element-6-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="(sideadjacenttoA)hypotenuse">(sideadjacentto∠A)hypotenuse(sideadjacentto∠A)hypotenuse= span id="MathJax-Element-7-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="ABAC">ABACABAC
• tan A = tangent of span id="MathJax-Element-8-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="A">∠A∠A = span id="MathJax-Element-9-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="(sideoppositetoA)(sideadjacenttoA)">(sideoppositeto∠A)(sideadjacentto∠A)(sideoppositeto∠A)(sideadjacentto∠A)= span id="MathJax-Element-10-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="BCAC">BCACBCAC = span id="MathJax-Element-11-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="sinAcosA">sinAcosAsinAcosA
• cosec A = cosecant of span id="MathJax-Element-12-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="A">∠A∠A = span id="MathJax-Element-13-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1sinA">1sinA1sinA
• sec A = secant of span id="MathJax-Element-14-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="A">∠A∠A = span id="MathJax-Element-15-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1cosA">1cosA1cosA
• cot A = cotangent  of span id="MathJax-Element-16-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="A">∠A∠A = span id="MathJax-Element-17-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1tanA">1tanA

2. Trigonometric Ratios of Specific Angles

The trigonometric ratios of some special angles, i.e., 0°, 30°, 45°, 60° and 90° follow a pattern and are easy to remember. Identifying and remembering this pattern helps in solving problems involving these angles.

Two angles are said to be complementary if their sum is 90°. Thus θ and (90° – θ) are complementary angles. Trigonometric ratios of complementary angles help in simplifying problems. Representing complementary angles in terms of these standard angles helps in solving a complex problem involving trigonometric ratios.

Trigonometric ratios of specific angles

Trigonometric ratios of complementary angles

• sin (span id="MathJax-Element-18-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="90">90∘90∘ – A) = cos A
  • cos (span id="MathJax-Element-19-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="90">90∘90∘ – A) = sin A
• tan (span id="MathJax-Element-20-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="90">90∘90∘ – A) = cot A
  • cot (span id="MathJax-Element-21-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="90">90∘90∘ – A) = tan A
• sec (span id="MathJax-Element-22-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="90">90∘90∘ – A) = cosec A
  • cosec (span id="MathJax-Element-23-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="90">90∘90∘ – A) = sec A

 

3. Trigonometric Identities.

In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem "a² +b² = c²" for right triangles or the identity (a + b)² = a² + b² + 2ab.

These identities are useful whenever expressions involving trigonometric functions need to be simplified.

A trigonometric identity is an equation involving trigonometric ratios of an angle, where the equation holds true for a defined range of values of the angle.

Fundamental trigonometric identities for all 0 span id="MathJax-Element-24-Frame" class="MathJx" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="">≤θ≤≤θ≤ 90

• span id="MathJax-Element-25-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="(sin)2+(cos)2=1">(sinθ)2+(cosθ)2=1(sinθ)2+(cosθ)2=1
• span id="MathJax-Element-26-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1+(tan)2=(sec)2">1+(tanθ)2=(secθ)21+(tanθ)2=(secθ)2
• span id="MathJax-Element-27-Frame" class="MathJax" style="box-sizing: border-box; margin: 0px; padding: 0px; word-break: break-word; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 1em; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; position: relative;" tabindex="0" data-mathml="1+(cot)2=(cosec)2">1+(cotθ)2=(cosecθ)2