In trigonometry, we write cos 30\u00b0 mathematically, and its exact value in fraction form is \u221a3\/2. Therefore, we write it in the following form in trigonometry.<\/span><\/p>\n cos (30\u00b0) = cos \u03c0\/6 = \u221a3\/2<\/b><\/p>\n <\/p>\n The exact cosine value of angle 30 degrees is \u221a3\/2 equal to 0.8660254037\u2026 in decimal form. The approximate value of the cosine of angle 30 is equal to 0.8660.<\/span><\/p>\n Cos (30\u00b0) = 0.8660254037\u2026 \u2248 0.8660<\/b><\/p>\n <\/b><\/p>\n The exact value of cos \u03c0\/6 can be derived using three methods explained below.<\/span><\/p>\n <\/span><\/p>\n We must know the relation between the sides of a right triangle when one of its angles is 30\u00b0. According to this property, the length of the opposite side is half of the hypotenuse length (<\/span>to angle 30\u00b0,i.e, the perpendicular<\/span>)<\/span>.<\/span><\/p>\n We will evaluate the exact value of the cosine of angle 30 degrees by using this property.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n In this case, we are considering the length of the hypotenuse as \u2018d\u2019, then the length of the opposite side (<\/span>to angle 30\u00b0,i.e, the perpendicular), PQ<\/span> will be \u2018d\/2\u2019.<\/span><\/span><\/p>\n Now, the lengths of the opposite side and hypotenuse are known, but the length of the adjacent side <\/span>(to angle 30\u00b0)<\/span>, <\/span>OQ<\/span> is unknown.<\/span><\/p>\n It is essential to find the value of cos (30\u00b0).\u00a0<\/span><\/p>\n So, apply the Pythagorean Theorem to find the value of the adjacent side.<\/span><\/span><\/p>\n \\text { Hypotenuse }^{2}=\\text { Perpendicular }^{2}+\\text { Adjacent Side }^{2}<\/span><\/span><\/p>\n \\mathrm{OP}^{2}=\\mathrm{OQ}^{2}+\\mathrm{PQ}^{2} <\/span> Here d represents the length of hypotenuse.<\/span><\/p>\n \u21d2 Length of adjacent side\/Hypotenuse = \u221a3\/2<\/span>\u00a0<\/span><\/p>\n Therefore, we can write that <\/span>cos (30\u00b0) = \u221a3\/2<\/b><\/p>\n <\/span><\/p>\n We can find the value of the cosine of angle 30\u00b0 by constructing a right-angled triangle with a 30\u00b0 angle by using geometrical tools.<\/span><\/span><\/p>\n Draw a straight horizontal line from Point G and then construct an angle of 30\u00b0 using the protractor.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Set compass to any length by a ruler. Here, the compass is set to 7.5 cm. Now, draw an arc on the 30\u00b0 angle line from point G, and it intersects the line at point H.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Finally, draw a perpendicular line on the horizontal line from point H, and it intersects the horizontal line at point I perpendicularly. Thus, a right-angled triangle \u2206HGI is formed.\u00a0<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Now,\u00a0 calculate the value of the cosine of 30 degrees and for this, measure the length of the adjacent side by a ruler. You will observe that the length of the adjacent side is 6.5 cm. The length of the hypotenuse is taken as 7.5 cm in this example.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Now, find the ratio of lengths of the adjacent side to the hypotenuse and get the value of the cosine of angle 30\u00b0.<\/span><\/p>\n cos (30\u00b0) = GI\/GH = 6.5\/7.5<\/span><\/p>\n So, cos (30\u00b0) = 0.866666\u2026 \u2248 0.8660<\/span><\/p>\n <\/p>\n We can prove the value of cos (30\u00b0) with a trigonometric approach.<\/span><\/p>\n we know that, sin 30\u00b0 = 1\/2<\/span><\/p>\n Also, by trigonometric identities,<\/span><\/p>\n \\sin ^{2} x+\\cos ^{2} x=1 <\/span> Put x = 30\u00b0<\/span><\/p>\n \\cos ^{2} 30^{\\circ}=1-\\sin ^{2} 30^{\\circ}<\/span><\/span><\/p>\n Put the value of sin 30\u00b0\u00a0<\/span><\/p>\n \\cos ^{2} 30^{\\circ}=1-(1 \/ 2)^{2}<\/span><\/span><\/p>\n\\cos ^{2} 30^{\\circ}=1-1 \/ 4 <\/span>\n \\cos ^{2} 30^{\\circ}=3 \/ 4 <\/span> \u00a0Hence, we proved the value of cos (30\u00b0) using different approaches.<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n 1. Evaluate: cos 30\u00b0 + sin 60\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n We know that cos (30\u00b0) = sin (60\u00b0) = \u221a3\/2 <\/p>\n 2. Evaluate: 2 cos 30\u00b0 \u2013 2 sin 30\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n We know that cos (30\u00b0) = \u221a3\/2, [\/et_pb_text][\/et_pb_column][et_pb_column type=”2_5″ module_id=”stickysideR” admin_label=”Column R” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fdefe0″ custom_padding=”25px|25px|25px|25px|true|true” sticky_position=”top” sticky_offset_top=”-280px” sticky_limit_top=”row” sticky_limit_bottom=”row” sticky_position_tablet=”none” sticky_position_phone=”none” sticky_position_last_edited=”on|desktop” sticky_limit_bottom_tablet=”” sticky_limit_bottom_phone=”” sticky_limit_bottom_last_edited=”on|phone” border_radii=”on|15px|15px|15px|15px” box_shadow_style=”preset3″ global_colors_info=”{}”][et_pb_image src=”https:\/\/eistudymaterial.s3.amazonaws.com\/1080×1080.png” alt=”Free Trial banner” title_text=”Mindspark Free Trial Banner” url=”https:\/\/mindspark.in\/free-trial” align=”center” module_class=”adsimg” _builder_version=”4.11.1″ _module_preset=”default” custom_padding=”||||false|false” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}” transform_styles__hover_enabled=”on|hover” transform_scale__hover_enabled=”on|hover” transform_translate__hover_enabled=”on|desktop” transform_rotate__hover_enabled=”on|desktop” transform_skew__hover_enabled=”on|desktop” transform_origin__hover_enabled=”on|desktop” transform_scale__hover=”102%|102%”][\/et_pb_image][et_pb_text admin_label=”Explore Other Topics [\/et_pb_text][et_pb_text _builder_version=”4.10.7″ _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” locked=”off” global_colors_info=”{}”]<\/p>\n [\/et_pb_text][et_pb_text admin_label=”Related Concepts [\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _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” locked=”off” global_colors_info=”{}”]<\/p>\nValue of Cos 30\u00b0<\/b><\/h2>\n
Proof<\/b><\/h2>\n
\n
Theoretical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-30-Degrees-01-300x200.png\")
<\/span><\/p>\n
\\Rightarrow \\mathrm{d}^{2}=\\mathrm{OQ}^{2}+(\\mathrm{d} \/ 2)^{2} <\/span>
\\Rightarrow \\mathrm{d}^{2}=\\mathrm{OQ}^{2}+\\mathrm{d}^{2} \/ 4 <\/span>
\\Rightarrow \\mathrm{OQ}^{2}=\\mathrm{d}^{2}-\\mathrm{d}^{2} \/ 4 <\/span>
\\Rightarrow \\mathrm{OQ}^{2}=(3 \/ 4) \\mathrm{d}^{2} <\/span>
\\Rightarrow \\mathrm{OQ}=(\\sqrt{3} \/ 2) \\mathrm{d} <\/span>
\\Rightarrow \\mathrm{OQ} \/ \\mathrm{d}=\\sqrt{3} \/ 2<\/span><\/span><\/p>\n\n
Practical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-30-Degrees-02-300x165.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-30-Degrees-03-300x176.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-30-Degrees-04-300x184.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-30-Degrees-05-300x190.png\")
<\/span><\/p>\n\n
Trigonometric approach<\/b><\/h3>\n<\/li>\n<\/ul>\n
\\text { Or } \\cos ^{2} x=1-\\sin ^{2} x<\/span>
<\/span><\/p>\n
\\cos \\left(30^{\\circ}\\right)=\\sqrt{(3 \/ 4)}=\\sqrt{3} \/ 2<\/span><\/p>\nExample<\/b><\/h2>\n
So, cos (30\u00b0) + sin (60\u00b0)
= \u221a3\/2 + \u221a3\/2
= 2(\u221a3\/2)
= \u221a3
<\/b><\/p>\n
and sin (30\u00b0) = 1\/2
So, 2 cos (30\u00b0) \u2013 2 sin (30\u00b0)
= 2 (\u221a3\/2) – 2(1\/2)
= \u221a3 – 1<\/p>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nExplore Other Topics<\/h1>\n
\n” _builder_version=”4.9.11″ _module_preset=”default” header_font=”|700|||||||” header_font_size=”25px” text_orientation=”center” custom_margin=”0px||0px||true|false” custom_padding=”8px|15px|0px|15px|false|true” locked=”off” global_colors_info=”{}”]<\/p>\nRelated Concepts<\/h1>\n