In trigonometry, we write tan (30\u00b0) mathematically, and its exact value in fraction form is 1\/\u221a3. Therefore, we write it in the following form in trigonometry.<\/span><\/p>\n tan (30\u00b0) = tan \u03c0\/6 = 1\/\u221a3<\/b><\/p>\n <\/p>\n The exact value of tan \u03c0\/6 is 1\/\u221a3 equal to 0.5773502691\u2026 in decimal form. It is reciprocal of cot 30 degrees. The approximate value of the tangent of angle 30\u00b0 is equal to 0.57735.<\/span><\/p>\n tan (30\u00b0) = 0.5773502691\u2026 \u2248 0.57735<\/b><\/p>\n <\/p>\n The exact value of tan \u03c0\/6 can be derived using three methods explained below.<\/span><\/p>\n <\/p>\n The exact value of tan (30\u00b0) can be derived on the basis of geometrical relations between sides of the right-angled triangle when one of the angles of the triangle is 30 degrees.<\/span><\/p>\n According to the properties of a right-angled triangle, if one of the angles of the right triangle is 30\u00b0, then the length of the adjacent side to 30\u00b0 is \u221a3\/2 times the length of the hypotenuse.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Therefore, in <\/span>\u25b3OPQ,\u00a0<\/span><\/p>\n OQ = (\u221a3\/2) \u00d7 OP<\/span><\/p>\n or OP = (2\/\u221a3) \u00d7 OQ<\/span><\/p>\n Now, apply the Pythagorean Theorem:\u00a0<\/span><\/p>\n \\text { Hypotenuse }^{2}=\\text { Perpendicular }^{2}+\\text { Adjacent } \\text { Side }^{2}<\/span><\/span><\/p>\n \\mathrm{OP}^{2}=\\mathrm{OQ}^{2}+\\mathrm{PQ}^{2}<\/span><\/span><\/p>\n Substituting the value of OP in terms of OQ<\/span><\/p>\n \\Rightarrow(2 \/ \\sqrt{3})^{2} \\times \\mathrm{OQ}^{2}=\\mathrm{OQ}^{2}+\\mathrm{PQ}^{2} <\/span> <\/p>\n PQ and OQ are lengths of opposite and adjacent sides of the right-angled triangle.<\/span><\/p>\n \u21d2 Length of opposite side\/adjacent side = 1\/\u221a3<\/span><\/p>\n The angle of <\/span>\u25b3OPQ is <\/span>30\u00b0.<\/span><\/p>\n Therefore, we can write that <\/span>tan (30\u00b0) = 1\/\u221a3<\/b>.<\/b><\/p>\n <\/b><\/p>\n You can also find the value of the tangent of angle 30\u00b0 practically by constructing a right-angled triangle with a 30\u00b0 angle by geometrical tools.<\/span><\/b><\/p>\n Draw a straight horizontal line from Point H 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 cm. Now, draw an arc on the 30\u00b0 angle line from point H, and it intersects the line at point I.<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n Finally, draw a perpendicular line on the horizontal line from 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 Now,\u00a0 calculate the value of the tangent of 30 degrees and for this, measure the length of the adjacent side (HJ) with a ruler. You will observe that the length of the opposite side (IJ) is 3.5 cm, and the length of the adjacent side is 6.05 cm in this example.<\/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 30\u00b0.<\/span><\/p>\n tan (30\u00b0) = IJ\/GJ = (3.5)\/(6.05)<\/span><\/p>\n So, tan (30\u00b0) = 0.578512396\u2026 \u2248 0.57735<\/span><\/p>\n <\/span><\/p>\n <\/span><\/p>\n We can prove the value of tan (30\u00b0) with a trigonometric approach.<\/span><\/p>\n we know that sin 30\u00b0 = (\u00bd) and cos 30\u00b0 = (\u221a3\/2)<\/span><\/p>\n Also, by trigonometric identities,<\/span><\/p>\n sin x\/cos x = tan x<\/b><\/p>\n Put x = 30\u00b0<\/span><\/p>\n tan (30\u00b0) = sin (30\u00b0)\/cos (30\u00b0)<\/span><\/p>\n Substitute the values of sin 30\u00b0 and cos 30\u00b0<\/p>\n tan (30\u00b0) = (\u00bd)\/(\u221a3\/2)<\/span><\/p>\n tan (30\u00b0) = 1\/\u221a3<\/span><\/p>\n Hence, we proved the value of tan (30\u00b0) using different approaches.<\/p>\n <\/span><\/p>\n 1. Evaluate: tan 30\u00b0 + sin 30\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n <\/strong> We know that tan (30\u00b0) = 1\/\u221a3 and sin (30\u00b0) = 1\/2 <\/p>\n 2. Evaluate: <\/strong>2 tan 30\u00b0 \u2013 2 cos 30\u00b0<\/strong><\/p>\n Solution:<\/strong><\/p>\n We know that tan (30\u00b0) = 1\/\u221a3 and cos (30\u00b0) = \u221a3\/2 <\/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>\nValue of Tan 30\u00b0<\/b><\/h2>\n
Proof<\/b><\/h2>\n
\n
Theoretical Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-30-Degrees-01-300x200.png\")
<\/span><\/p>\n
\\Rightarrow(4 \/ 3) \\mathrm{OQ}^{2}=\\mathrm{OQ}^{2}+\\mathrm{PQ}^{2} <\/span>
\\Rightarrow(4 \/ 3) \\mathrm{OQ}^{2}-\\mathrm{OQ}^{2}=\\mathrm{PQ}^{2} <\/span>
\\Rightarrow(1 \/ 3) \\mathrm{OQ}^{2}=\\mathrm{PQ}^{2} <\/span>
\\Rightarrow \\mathrm{PQ}^{2} \/ \\mathrm{OQ}^{2}=1 \/ 3 <\/span>
\\Rightarrow \\mathrm{PQ} \/ \\mathrm{OQ}=1 \/ \\sqrt{3}<\/span><\/span><\/p>\n\n
Practical Method<\/b><\/b><\/h3>\n<\/li>\n<\/ul>\n
![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-30-Degrees-02-300x165.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-30-Degrees-03-300x176.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-30-Degrees-04-300x186.png\")
<\/span><\/p>\n![\"\"](\"https:\/\/eistudymaterial.s3.amazonaws.com\/Tan-30-Degrees-05-300x190.png\")
<\/span><\/p>\n\n
Trigonometric Method<\/b><\/h3>\n<\/li>\n<\/ul>\n
<\/b><\/h2>\n
Example<\/b><\/h2>\n
So, tan (30\u00b0) + sin (30\u00b0)
= 1\/\u221a3 + 1\/2
= (2+\u221a3)\/2\u221a3<\/p>\n
So, 2 tan (30\u00b0) \u2013 2 cos (30\u00b0)
= 2 (1\/\u221a3) – 2(\u221a3\/2)
= 2\/\u221a3 – \u221a3
= -(1\/\u221a3)<\/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