In trigonometry, we write the exact value of cos 60\u00b0 mathematically. Its exact value in fraction form is \u00bd equal to 0.5 in the decimal form. Therefore, we write it in the following form in trigonometry.<\/span><\/p>\n cos (60\u00b0) = cos \u03c0\/3 = \u00bd = 0.5<\/b><\/p>\n <\/b><\/p>\n The exact value of cos 60 degree can be derived using three methods explained below.<\/span><\/p>\n <\/span><\/p>\n To find the value of the cosine of angle 60 degrees, let us consider an equilateral triangle given below:<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Since all sides of an equilateral triangle are equal, AB = BC = AC, and AD is perpendicular bisector, dividing BC into two equal parts.<\/span><\/span><\/p>\n Let us consider the length of each side of the triangle as 2 units.<\/span><\/p>\n That is AB = AC = BC = 2 units and CD = BD = 2\/2 = 1 unit.<\/span><\/p>\n In the <\/span>\u25b3ABC<\/span>,<\/span><\/p>\n The value of cos (60\u00b0) = adjacent side\/hypotenuse = BD\/AB = \u00bd<\/span><\/span><\/span><\/p>\n Similarly, we can determine the value of sin 60\u00b0 by evaluating the required sides.<\/span><\/span><\/p>\n In right triangle ABD, Using the Pythagoras theorem:<\/span><\/p>\n AB<\/span>\u00b2<\/span> = AD\u00b2<\/span>\u00a0<\/span>+ BD<\/span>\u00b2<\/span><\/p>\n 2<\/span>\u00b2<\/span> = AD<\/span>\u00b2<\/span> + 1<\/span>\u00b2<\/span><\/p>\n AD<\/span>\u00b2<\/span> = 2<\/span>\u00b2<\/span> -1<\/span>\u00b2<\/span><\/p>\n AD<\/span>\u00b2<\/span> = 4 \u2013 1<\/span><\/p>\n AD<\/span>\u00b2<\/span> = 3<\/span><\/p>\n AD = \u221a3<\/span><\/p>\n Now,\u00a0<\/span><\/p>\n Sin 60\u00b0 = opposite side\/hypotenuse = AD\/AB = \u221a3\/2<\/span><\/p>\n <\/p>\n You can also find the value of cos of angle 60\u00b0 practically by constructing a right-angled triangle with 60\u00b0 angle by geometrical tools.<\/span><\/p>\n Draw a straight horizontal line from Point A and then construct an angle of 60\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 4.5 cm. Now, draw an arc on the 45\u00b0 angle line from point A.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Finally, draw a perpendicular line on the horizontal line from point D, and it intersects the horizontal line at point E perpendicularly. Thus, a right-angled triangle \u2206ADE is formed.\u00a0<\/span><\/p>\n Now, calculate the value of the cosine of 60 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 2.3 cm. The length of the hypotenuse is taken as 4.5 cm in this example.<\/span><\/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 60\u00b0.<\/span><\/p>\n cos (45\u00b0) = AE\/AD = (2.3)\/(4.5)<\/span><\/p>\n So, cos (45\u00b0) = 0.5111\u2026 \u2248 0.5<\/span><\/p>\n <\/p>\n We can prove the value of cos (60\u00b0) with a trigonometric approach.<\/span><\/p>\n We know that Sin 60\u00b0 = \u221a3\/2<\/span><\/p>\n Also, by trigonometric identities,<\/span><\/p>\n \\sin ^{2} x+\\cos ^{2} x=1<\/span> Put x = 60\u00b0<\/span><\/p>\n \\cos ^{2} 60^{\\circ}=1-\\sin ^{2} 60^{\\circ}<\/span><\/span><\/p>\n <\/span><\/p>\n Put the value of sin 60\u00b0\u00a0<\/span><\/p>\n \\cos ^{2} 60^{\\circ}=1-(\\sqrt{3} \/ 2)^{2} <\/span> cos (60\u00b0) = \\sqrt{1\/4}<\/span> = 1\/2<\/span><\/span><\/p>\n Hence, we proved the value of cos (60\u00b0) using different approaches.<\/span><\/p>\n <\/span><\/p>\n <\/b><\/p>\n 1. Evaluate: cos 60\u00b0 + sin 30\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n We know that cos (60\u00b0) = sin (30\u00b0) = 1\/2. \u00a0<\/span><\/p>\n 2. Evaluate: 2 sin 60\u00b0 \u2013 4 cos 60\u00b0<\/strong><\/p>\n Solution:<\/strong> We know that cos (60\u00b0) = \u00bd and sin (60\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>\nProof<\/b><\/h2>\n
\n
Theoretical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-60-Degrees-01-300x275.png\")
<\/span><\/p>\n\n
Practical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-60-Degrees-02-300x288.png\")
‘<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-60-Degrees-04-300x288.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-60-Degrees-03-1-300x296.png\")
<\/span><\/p>\n\n
Trigonometric Method<\/b><\/span><\/h3>\n<\/li>\n<\/ul>\n
Or \\cos ^{2} x=1-\\sin ^{2} x<\/span><\/span><\/p>\n
\\cos ^{2} 60^{\\circ}=1-3 \/ 4<\/span>
<\/span><\/p>\nExample<\/b><\/h2>\n
So, cos (60\u00b0) + sin (30\u00b0)
= 1\/2 + 1\/2
= 1<\/p>\n
<\/span><\/p>\n
So, 2 sin (60\u00b0) \u2013 4 cos (60\u00b0)
= 2 (\u221a3\/2) – 4(1\/2)
= \u221a3 – 2
<\/span><\/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