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The trigonometric function \\sin 3x<\/span> formula equals \\left(3 \\sin x-4 \\sin ^{3} x\\right)<\/span>.<\/span>In trigonometry, \\sin 3x<\/span> is a triple angle identity. We can derive this formula using the angle addition identity of the sin function.\u00a0<\/span><\/p>\n <\/p>\n For deriving the formula, we will write the angle 3x as (2x + x). Apart from this, we will use a few trigonometric identities given below:<\/span><\/p>\n <\/span><\/p>\n Now, let\u2019s derive the formula using the above details.<\/span><\/p>\n \\sin 3x=\\sin(2x+x)<\/span><\/span><\/span><\/p>\n =(\\sin 2x\\cos x)+(\\cos 2x\\sin x) \\text { [Using } \\sin(a+b) \\text { formula]}<\/span><\/span><\/p>\n =(2 \\sin x \\cos x) \\cos x+\\left(1-2 \\sin ^{2} x\\right) \\sin x \\text { [Using } \\sin 2 x \\text { and } \\cos 2 x \\text { formula] }<\/span><\/p>\n =\\left(2 \\cos ^{2} x \\sin x\\right)+(\\sin x)-\\left(2 \\sin ^{3} x\\right)<\/span><\/p>\n =2\\left(1-\\sin ^{2} x\\right) \\sin x+(\\sin x)-\\left(2 \\sin ^{3} x\\right)\\left[\\text { Since } \\sin ^{2} x+\\cos ^{2} x=1, \\text { Hence } \\cos ^{2} x=1-\\sin ^{2} x\\right]<\/span><\/p>\n =2 \\sin x-2 \\sin ^{3} x+(\\sin x)-\\left(2 \\sin ^{3} x\\right)<\/span><\/p>\n =(2 \\sin x+\\sin x)-2 \\sin ^{3} x-2 \\sin ^{3} x<\/span><\/p>\n =(2 \\sin x+\\sin x)-\\left(2 \\sin ^{3} x+2 \\sin ^{3} x\\right)<\/span><\/p>\n =\\left(3 \\sin x-4 \\sin ^{3} x\\right)<\/span><\/p>\n Thus we proved the formula of \\sin 3x<\/span> using the angle addition identity.<\/span><\/p>\n <\/p>\n Assume 3x = 180\u00b0\u00a0<\/span><\/p>\n \u21d2 x = 180\u00b0\/3 = 60\u00b0<\/span><\/p>\n <\/span><\/p>\n We know that the formula of \\sin 3 x=3 \\sin x-4 \\sin ^{3} x<\/span><\/span><\/p>\n Substituting the values of 3x and x\u00a0<\/span><\/p>\n\\operatorname{Sin} 180^{\\circ}=3 \\sin 60^{\\circ}-4 \\sin ^{3}\\left(60^{\\circ}\\right)<\/span>\n=3(\\sqrt{3} \/ 2)-4(\\sqrt{3} \/ 2)^{3}<\/span>\n = 3(\u221a3\/2) – 4 {(3(\u221a3)\/8}<\/span><\/p>\n = 3(\u221a3\/2) – 3(\u221a3\/2)<\/span><\/p>\n = 0<\/span><\/p>\n Therefore, the value of sin 180\u00b0 is 0 using the sin 3x identity.<\/span><\/p>\n <\/span><\/p>\n We know that the formula of \\sin 3 x=3 \\sin x-4 \\sin ^{3} x<\/span>.<\/span><\/p>\n Assume 3x = 270\u00b0\u00a0<\/span><\/p>\n \u21d2 x = 270\u00b0\/3 = 90\u00b0<\/span><\/p>\n Substituting the values of 3x and x\u00a0<\/span><\/p>\n \\sin 270^{\\circ}=3 \\sin \\left(90^{\\circ}\\right)-4 \\sin ^{3}\\left(90^{\\circ}\\right)<\/span><\/span><\/p>\n =(3 \\times 1)-4 \\times(1)^{3}<\/span><\/span><\/p>\n = 3 – 4<\/span><\/p>\n = (-1)<\/span><\/p>\n <\/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 [\/et_pb_text][\/et_pb_column][\/et_pb_row][et_pb_row admin_label=”Row for space” _builder_version=”4.10.6″ _module_preset=”default” locked=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.11″ _module_preset=”default” global_colors_info=”{}”][et_pb_divider show_divider=”off” _builder_version=”4.10.4″ _module_preset=”default” global_colors_info=”{}”][\/et_pb_divider][\/et_pb_column][\/et_pb_row][\/et_pb_section][et_pb_section fb_built=”1″ admin_label=”banner and faq Section” module_class=”mainsec2″ _builder_version=”4.10.4″ _module_preset=”default” custom_padding=”40px||0px||false|false” global_colors_info=”{}”][et_pb_row use_custom_gutter=”on” gutter_width=”1″ make_equal=”on” disabled_on=”on|on|off” admin_label=”banner Row” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#fff7d6″ width=”100%” max_width=”1310px” height=”134px” custom_margin=”||50px||false|false” custom_padding=”12px||12px||true|false” border_radii=”on|11px|11px|11px|11px” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_image src=”https:\/\/stgwebsite.mindspark.in\/wordpress\/wp-content\/uploads\/2021\/08\/calloutImage.png” title_text=”calloutImage” show_bottom_space=”off” admin_label=”Image” module_class=”img1″ _builder_version=”4.10.8″ _module_preset=”default” width=”25px” height=”60px” custom_padding=”2px||2px||true|false” global_colors_info=”{}”][\/et_pb_image][et_pb_text module_class=”ftstyle” _builder_version=”4.9.10″ _module_preset=”default” text_orientation=”center” global_colors_info=”{}”]<\/p>\nDerivation of Sin 3x Formula<\/b><\/h2>\n
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Examples\u00a0<\/b><\/h2>\n
1. Determine the value of sin 180\u00b0 using the sin 3x identity?<\/span><\/h3>\n
2. Calculate the value of sin 270\u00b0 using the sin 3x formula<\/span><\/h3>\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