{"id":5979,"date":"2021-12-16T04:06:36","date_gmt":"2021-12-16T04:06:36","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=5979"},"modified":"2021-12-29T11:11:48","modified_gmt":"2021-12-29T11:11:48","slug":"trigonometry-formulas-with-examples-and-faqs","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/trigonometry-formulas-with-examples-and-faqs\/","title":{"rendered":"Trigonometry Formulas with Examples and FAQs"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
\n” _builder_version=”4.11.3″ _module_preset=”default” header_font=”|700|||||||” header_text_align=”left” header_font_size=”50px” header_line_height=”1.18em” custom_padding=”|0px||4px|false|false” header_font_size_tablet=”” header_font_size_phone=”35px” header_font_size_last_edited=”on|phone” global_colors_info=”{}”]<\/p>\n

Trigonometry Formulas with Examples and FAQs<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

Trigonometry Formulas<\/b><\/b><\/h2>\n

Trigonometry finds its use in establishing relations between angles, lengths of the sides of any triangle and height of any triangle. We generally associate it with right-angled triangles. One such triangle is shown below, to establish relations between the angle <\/span>\u03b8<\/span> and the sides, i.e., perpendicular(the side opposite to <\/span>\u03b8<\/span>), base(the side adjacent to <\/span>\u03b8)<\/span> and hypotenuse(the side opposite to 90\u00b0) in case of a right-angled triangle.<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

Basic Trigonometric Formulas<\/b><\/h3>\n

The basic trigonometric ratios are namely sine(sin), cosine(cos), tangent(tan), cosecant(cosec), secant(sec) and cotangent(cot). The trigonometry ratios for the angle <\/span>\u03b8<\/span> with respect to the above figure is:<\/span><\/p>\n

1. \\sin \\theta=\\frac{\\text { perpendicular }}{\\text { hypotemuse }}<\/span>
<\/span><\/p>\n

2. \\cos \\theta=\\frac{\\text { base }}{\\text { hypotenuse }}<\/span>
<\/span><\/p>\n

3. \\tan \\theta=\\frac{\\text { perpendicular }}{\\text { base }}<\/span>
<\/span><\/p>\n

4. \\operatorname{cosec} \\theta=\\frac{\\text { hypotemuse }}{\\text { perpendicular }}<\/span>
<\/span><\/p>\n

5. \\sec \\theta=\\frac{\\text { hypotenuse }}{\\text { base }}<\/span>
<\/span><\/p>\n

6. \\cot \\theta=\\frac{\\text { base }}{\\text { perpendicular }}<\/span><\/span><\/p>\n

<\/span><\/p>\n

Reciprocal Relations between Trigonometric Ratios<\/b><\/h3>\n

1. \\operatorname{cosec} \\theta=\\frac{1}{\\sin \\theta}<\/span>
<\/b><\/p>\n

2. \\sec \\theta=\\frac{1}{\\cos \\theta}<\/span>
<\/b><\/p>\n

3. \\cot \\theta=\\frac{1}{\\tan \\theta}<\/span><\/b><\/p>\n

<\/b><\/p>\n

Relations between Trigonometric Ratios<\/b><\/h3>\n

1. \\tan \\theta=\\frac{\\sin \\theta}{\\cos \\theta}<\/span>
<\/b><\/p>\n

2. \\cot \\theta=\\frac{\\cos \\theta}{\\sin \\theta}<\/span><\/b><\/p>\n

<\/b><\/p>\n

Trigonometric Ratios of Complementary Angles<\/b><\/b><\/h3>\n

1. \\sin \\left(90^{\\circ}-\\theta\\right)=\\cos \\theta<\/span>
<\/b><\/p>\n

2. \\cos \\left(90^{\\circ}-\\theta\\right)=\\sin \\theta<\/span>
<\/b><\/p>\n

3. \\tan \\left(90^{\\circ}-\\theta\\right)=\\cot \\theta<\/span>
<\/b><\/p>\n

4. \\operatorname{cosec}\\left(90^{\\circ}-\\theta\\right)=\\sec \\theta<\/span>
<\/b><\/p>\n

5. \\sec \\left(90^{\\circ}-\\theta\\right)=\\operatorname{cosec} \\theta<\/span>
<\/b><\/p>\n

6. \\cot \\left(90^{\\circ}-\\theta\\right)=\\tan \\theta<\/span><\/b><\/p>\n

<\/b><\/p>\n

Table for Values of Trigonometric Ratios for standard angles (0<\/b>\u00b0<\/b> to 90<\/b>\u00b0<\/b>)<\/b><\/h3>\n

\"\"<\/b><\/p>\n

<\/b><\/p>\n

Trigonometric Identities<\/b><\/h3>\n

1. \\sin ^{2} \\theta+\\cos ^{2} \\theta=1<\/span>
<\/b><\/p>\n

2. \\tan ^{2} \\theta+1=\\sec ^{2} \\theta<\/span>
<\/b><\/p>\n

3. \\cot ^{2} \\theta+1=\\operatorname{cosec}^{2} \\theta<\/span><\/b><\/p>\n

<\/b><\/p>\n

Examples<\/b><\/h2>\n

Example 1:<\/strong> If the value of \\sin \\theta=\\frac{12}{13}<\/span><\/span>, find all the remaining trigonometric ratios.<\/span><\/b><\/p>\n

Solution:<\/strong><\/p>\n

Given, the value of \\sin \\theta=\\frac{12}{13}<\/span>, <\/b><\/span><\/p>\n

We know, \\sin \\theta=\\frac{\\text { perpendicular }}{\\text { hypotenuse }}<\/span><\/span><\/p>\n

Perpendicular = 12 units and Hypotenuse = 13 units.<\/span><\/p>\n

By Pythagoras theorem for right triangle we have,<\/span><\/p>\n

\\text { Perpendicular }^{2}+\\text { base }^{2}=\\text { hypotenuse }^{2}<\/span><\/span><\/p>\n

\\Rightarrow \\text { Base }=\\sqrt{\\text { Hypotenuse }^{2}-\\text { Perpendicular }^{2}}<\/span><\/span><\/p>\n

Substituting Value of perpendicular and hypotenuse we get:<\/span><\/p>\n

\\text { Base }=\\sqrt{13^{2}-12^{2}}=\\sqrt{169-144}=\\sqrt{25}=5 \\text { units }<\/span><\/span><\/p>\n

<\/span><\/p>\n

The trigonometry ratios for the angle <\/span>\u03b8<\/span> with respect to the given value of <\/span>sin<\/span> are:<\/span><\/p>\n

1. \\cos \\theta=\\frac{\\text { base }}{\\text { hypotenuse }}=\\frac{5}{13}<\/span>
2. \\tan \\theta=\\frac{\\text { perpendicular }}{\\text { base }}=\\frac{12}{5}<\/span>
3. \\operatorname{cosec} \\theta=\\frac{\\text { hypotense }}{\\text { perpendicular }}=\\frac{13}{12}<\/span>
4. \\sec \\theta=\\frac{\\text { hypotenuse }}{\\text { base }}=\\frac{13}{5}<\/span>
5. \\cot \\theta=\\frac{\\text { base }}{\\text { perpendicular }}=\\frac{5}{12}<\/span><\/span><\/p>\n

<\/span><\/p>\n

Example 2: Find the value of \\frac{\\sin 30^{\\circ}}{\\tan 45^{\\circ}}+\\cos ^{2}\\left(60^{\\circ}\\right)<\/span><\/span><\/p>\n

Solution<\/span>:\u00a0\u00a0<\/span><\/p>\n

\\sin 30^{\\circ}=\\frac{1}{2}, \\tan 45^{\\circ}=1 \\text { and } \\cos 60^{\\circ}=\\frac{1}{2}<\/span><\/span><\/p>\n


\\therefore\u00a0 \\frac{\\sin 30^{\\circ}}{\\tan 45^{\\circ}}+\\cos ^{2}\\left(60^{\\circ}\\right) <\/span>
=\\frac{\\frac{1}{2}}{1}+\\left(\\frac{1}{2}\\right)^{2} <\/span>
=\\frac{1}{2}+\\frac{1}{4} <\/span>
=\\frac{2+1}{4} <\/span>
=\\frac{3}{4}<\/span><\/span><\/p>\n

Hence, \\frac{\\sin 30^{\\circ}}{\\tan 45^{\\circ}}+\\cos ^{2}\\left(60^{\\circ}\\right)=\\frac{3}{4}<\/span><\/p>\n

<\/span><\/p>\n

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Explore Other Topics<\/h1>\n

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\n
Geometry<\/a><\/div>\n
Trigonometry<\/a><\/div>\n
Operations<\/a><\/div>\n
Numbers<\/a><\/div>\n<\/div>\n

[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

[\/et_pb_text][et_pb_text _builder_version=”4.9.11″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” hover_enabled=”0″ locked=”off” global_colors_info=”{}” theme_builder_area=”post_content” sticky_enabled=”0″]<\/p>\n

\n
Obtuse angle<\/a><\/div>\n
Reflex angle<\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1. Mention the basic trigonometric ratios?
    <\/strong><\/h3>\n

    Ans: <\/strong><\/span>The basic trigonometric ratios are:<\/span><\/p>\n

      \n
    • Sine (sin)<\/span><\/li>\n
    • Cosine (cos)<\/span><\/li>\n
    • Tangent (tan)<\/span><\/li>\n
    • Cosecant (cosec)<\/span><\/li>\n
    • Secant (sec)<\/span><\/li>\n
    • Cotangent (cot)<\/span><\/li>\n<\/ul>\n

      <\/strong>.
      <\/strong><\/span><\/span><\/p>\n

      Q2. What are the trigonometric identities?
      <\/strong><\/h3>\n

      Ans: <\/strong>The basic trigonometric identities are:
      <\/strong>
      <\/strong><\/p>\n

      1. \\sin ^{2} \\theta+\\cos ^{2} \\theta=1<\/span>
      2. \\tan ^{2} \\theta+1=\\sec ^{2} \\theta<\/span>
      3. \\cot ^{2} \\theta+1=\\operatorname{cosec}^{2} \\theta<\/span><\/p>\n

      [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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