The value of the tangent of the angle 45\u00b0 in the right-angled triangle is called tan of angle 45 degrees. The tangent of angle 45\u00b0 is a value representing the ratio of the opposite side\u2019s length to the adjacent side\u2019s length with respect to the considered angle.<\/span><\/p>\n <\/span><\/p>\n The exact value of tan (45\u00b0) is 1 in both fraction and decimal form.<\/span><\/p>\n tan (60\u00b0) = tan \u03c0\/4 = 1<\/b><\/p>\n <\/p>\n The exact value of tan \u03c0\/4 can be derived using three methods explained below.<\/span><\/p>\n <\/span><\/p>\n We can derive the exact value of tan (45\u00b0) by using the property of the right-angled triangle.\u00a0<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n According to the right-angled triangle property, if an angle of the triangle is 45\u00b0, the triangle forms an isosceles right triangle (as shown in the figure)<\/span><\/p>\n Take, the length of adjacent and the opposite sides to the angle QPR as \u2018l\u2019 and the length of the hypotenuse to the angle QPR as \u2018r\u2019.<\/span><\/p>\n tan \u03b8 = (Length of opposite side) \/ (Length of the adjacent side)<\/span><\/p>\n Here,<\/span><\/p>\n tan (45\u00b0) = QR\/PR<\/span><\/p>\n tan (45\u00b0) = l\/l = 1<\/span><\/p>\n Therefore, tan (45\u00b0) =1<\/span><\/p>\n <\/p>\n You can also find the value of the tangent of angle 45\u00b0 practically by constructing a right-angled triangle with a 45\u00b0 angle by geometrical tools.<\/span><\/strong><\/p>\n Draw a straight horizontal line from Point H and then construct an angle of 45\u00b0 using the protractor.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Set the compass to any length by a ruler. Here, the compass is set to 4.8 cm. Now, draw an arc on the 45\u00b0 angle line from point H, and it intersects the line at the point I.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Finally, draw a perpendicular line on the horizontal line from the point I, and it intersects the horizontal line at point J perpendicularly. Thus, a right-angled triangle \u2206HIJ is formed.\u00a0<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Now,\u00a0 calculate the value of the tangent of 45 degrees and for this, measure the length of the adjacent side (HJ) by a ruler. You will observe that the length of the opposite side (IJ) is 3.3 cm, and the length of the adjacent side is 3.37 cm in this example.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Now, find the ratio of lengths of the opposite side to the adjacent side and get the value of the tangent of angle 45\u00b0.<\/span><\/p>\n tan (45\u00b0) = IJ\/GJ = (3.3)\/(3.37)<\/span><\/p>\n So, tan (45\u00b0) = 0.97922\u2026 \u2248 1<\/span><\/p>\n <\/span><\/p>\n We can prove the value of tan (45\u00b0) with a trigonometric approach.<\/span><\/p>\n we know that sin 45\u00b0 = cos 45\u00b0 = 1\/\u221a2<\/span><\/p>\n Also, by trigonometric identities,<\/span><\/p>\n sin x\/cos x = tan x<\/b><\/p>\n Put x = 45\u00b0<\/span><\/p>\n tan (45\u00b0) = sin (45\u00b0)\/cos (45\u00b0)<\/span><\/p>\n Put the values of sin 45\u00b0 and cos 45\u00b0<\/span><\/p>\n tan (45\u00b0) = (1\/\u221a2)\/(1\/\u221a2)<\/span><\/p>\n tan (45\u00b0) = 1<\/span><\/p>\n Hence, we proved the value of tan (45\u00b0) using different approaches.<\/span><\/p>\n <\/p>\n <\/p>\n 1. Evaluate: tan 45\u00b0 + cos 60\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n We know that tan (45\u00b0) = 1 and cos (60\u00b0) = 1\/2 <\/p>\n 2. Evaluate: 2 tan 45\u00b0 \u2013 2 sin 30\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n We know that tan (45\u00b0) = 1 and sin (30\u00b0) = 1\/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 Tan 45\u00b0<\/b><\/h2>\n
Proof<\/b><\/h2>\n
\n
Theoretical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-45-Degrees-01.png\")
<\/span><\/p>\n\n
\u00a0Practical Method<\/strong><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-45-Degrees-02-300x230.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-45-Degrees-03-300x248.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-45-Degrees-04-300x259.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-45-Degrees-05-300x261.png\")
<\/span><\/p>\n\n
Trigonometric Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
Example<\/b><\/h2>\n
So, tan (45\u00b0) + cos (60\u00b0)
= 1 + 1\/2
= 3\/2
<\/b><\/p>\n
So, 2 tan (45\u00b0) \u2013 2 sin (30\u00b0)
= 2 – 2(1\/2)
= 2 – 1
= 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