<\/span><\/p>\n <\/span><\/p>\n In trigonometry, we write the cosine of angle 45\u00b0 mathematically, and its exact value in fraction form is 1\/\u221a2. Therefore, we write it in the following form in trigonometry.<\/span><\/p>\n cos (45\u00b0) = cos \u03c0\/4 = 1\/\u221a2<\/b><\/p>\n <\/p>\n The exact value of the cosine of angle 45 degrees is 1\/\u221a2 equal to 0.7071067812\u2026 in decimal form. The approximate value of the cosine of angle 45 is equal to 0.7071.<\/span><\/p>\n Cos (45\u00b0) = 0.7071067812\u2026 \u2248 0.7071<\/b><\/p>\n The exact value of cos 45 can be derived using three methods. We will use them one by one.<\/span><\/p>\n <\/span><\/p>\n According to the right-angled triangle property, the lengths of the sides adjacent and opposite to the angle \u03b8 are equal when the angle of the right-angled triangle is equal to 45\u00b0.\u00a0<\/span><\/p>\n Take, the length of both adjacent side (BC) and opposite side (AB) as \u2018l\u2019 (as the right-angled triangle is isosceles when one of its angles is 45\u00b0) and the length of hypotenuse as \u2018r\u2019. Now, according to the Pythagoras theorem, we know that\u00a0<\/span><\/p>\n \\text { Hypotenuse }^{2}=\\text { Perpendicular }^{2}+\\text { Adjacent Side }^{2}<\/span><\/span><\/p>\n \\text { So, } A C^{2}=A B^{2}+B C^{2}<\/span><\/span><\/p>\n \\Rightarrow \\mathrm{r}^{2}=\\mathrm{l}^{2}+\\mathrm{l}^{2} <\/span> \u21d2 Length of adjacent side\/Hypotenuse = 1\/\u221a2<\/span><\/p>\n Therefore, we can write that <\/span>cos (45\u00b0) = 1\/\u221a2<\/b><\/p>\n <\/b><\/p>\n You can also find the value of cos of angle 45\u00b0 practically by constructing a right-angled triangle with 45\u00b0 angle by geometrical tools.<\/span><\/span><\/p>\n Draw a straight horizontal line from Point I and then construct an angle of 45\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 6.5 cm. Now, draw an arc on the 45\u00b0 angle line from point I and it intersects the line at point J.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Finally, draw a perpendicular line on the horizontal line from point J and it intersects the horizontal line at point K perpendicularly. Thus, a right-angled triangle \u2206JIK is formed.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Now,\u00a0 calculate the value of the cosine of 45 and for this, measure the length of the adjacent side by a ruler. You will observe that the length of the adjacent side is 4.6 cm. The length of the hypotenuse is taken as 6.5 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 45\u00b0.<\/span><\/p>\n cos (45\u00b0) = IK\/IJ = 4.6\/6.5<\/span><\/p>\n So, cos (45\u00b0) = 0.7076923077\u2026 \u2248 0.7071<\/span><\/p>\n <\/p>\n We can prove the value of cos (45\u00b0) with a trigonometric approach.<\/span><\/p>\n we know that, sin 45\u00b0 = 1\/\u221a2<\/span><\/p>\n Also, by trigonometric identities,<\/span><\/p>\n \\sin ^{2} x+\\cos ^{2} x=1<\/span> Put the value of sin 45\u00b0\u00a0<\/span><\/p>\n \\cos ^{2} 45^{\\circ}=1-(1 \/ \\sqrt{2})^{2} <\/span> Hence, we proved the value of cos (45\u00b0) using different approaches.<\/span><\/p>\n <\/span><\/p>\n 1. Evaluate: cos 45\u00b0 + sin 45\u00b0<\/strong><\/b><\/span><\/p>\n We know that cos (45\u00b0) = sin (45\u00b0) = 1\/\u221a2. <\/p>\n 2. Evaluate: 2 sin 45\u00b0 \u2013 4 cos 45\u00b0<\/strong><\/p>\n We know that cos (45\u00b0) = sin (45\u00b0) = 1\/\u221a2. \u00a0<\/span><\/p>\n [\/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>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-45-Degrees-01-1-300x173.png\")
<\/span><\/p>\nValue of Cos 45\u00b0<\/b><\/h2>\n
<\/h2>\n
Proof<\/b><\/h2>\n
\n
Theoretical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/triangle-ee-300x264.png\")
<\/span><\/span><\/p>\n
\\Rightarrow \\mathrm{r}^{2}=2 \\mathrm{l}^{2} <\/span>
\\Rightarrow \\mathrm{r}=\\sqrt{2} \\mathrm{l} <\/span>
\\Rightarrow \\mathrm{l} \/ \\mathrm{r}=1 \/ \\sqrt{2}<\/span>
<\/span><\/p>\n\n
Practical Method<\/b><\/b><\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-45-Degrees-03-300x220.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-45-Degrees-04-300x219.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-45-Degrees-05-300x227.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Cos-45-Degrees-06-300x230.png\")
<\/span><\/p>\n\n
Trigonometric Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
Or \\cos ^{2} x=1-\\sin ^{2} x<\/span>
Put x=45^{\\circ}<\/span>
\\cos ^{2} 45^{\\circ}=1-\\sin ^{2} 45^{\\circ}<\/span><\/span><\/span><\/p>\n
\\cos ^{2} 45^{\\circ}=1-1 \/ 2 <\/span>
\\cos \\left(45^{\\circ}\\right)=\\sqrt{1 \/ 2}=1 \/ \\sqrt{2}<\/span>
<\/span><\/p>\nExample<\/b><\/b><\/h2>\n
So, cos (45\u00b0) + sin (45\u00b0)
= 1\/\u221a2 + 1\/\u221a2
= 2\/\u221a2
= \u221a2<\/p>\n
So, 2 sin (45\u00b0) \u2013 4 cos (45\u00b0)
= 2 (1\/\u221a2) – 4(1\/\u221a2)
= 2\/\u221a2 – 4\/\u221a2
= -2\/\u221a2
= -\u221a2<\/p>\n
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