{"id":6785,"date":"2021-12-27T06:44:59","date_gmt":"2021-12-27T06:44:59","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=6785"},"modified":"2022-01-02T14:05:25","modified_gmt":"2022-01-02T14:05:25","slug":"cos-90-degrees-value-of-cos-90-with-proof-examples-and-faq","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/cos-90-degrees-value-of-cos-90-with-proof-examples-and-faq\/","title":{"rendered":"Cos 90 Degrees: Value of cos 90 with Proof, Examples and FAQ"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
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Cos 90 Degrees: Value of cos 90 with Proof, Examples and FAQ<\/h1>\n

 <\/p>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

Cos 90 Degrees<\/b><\/h2>\n

In a right-angled triangle, the cosine function of an angle is the ratio of the length of the adjacent side and the hypotenuse side (of angle \u03b8).<\/span><\/p>\n

<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

In this article, we will discuss the cosine of angle 90 degrees value, which is equal to zero. We will also derive this value using the quadrants of a unit circle.<\/span><\/p>\n

\\cos \\left(90^{\\circ}\\right)=\\cos \\pi \/ 2=0<\/span><\/span><\/p>\n

<\/span><\/p>\n

Proof of\u00a0 Cos 90 Degrees<\/b><\/h2>\n

Now let us calculate the value of cos 90 using a unit circle that has its centre at the origin of the coordinate axes \u2018x\u2019 and \u2018y\u2019.\u00a0<\/span><\/p>\n

<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

Let P (a, b) be a point on the circle\u2019s circumference that forms an angle POA = x radian. It means that the length of the arc PA is equal to x. From this, we define the value of sin x = PM\/OP = b and cos x = OM\/OP = a.<\/span><\/span><\/p>\n

Consider a right-angled triangle OMP in the given unit circle.<\/span><\/p>\n

Using the Pythagoras theorem,<\/span><\/p>\n

\\mathrm{OM}^{2}+\\mathrm{MP}^{2}=\\mathrm{OP}^{2} <\/span>
\\Rightarrow \\mathrm{a}^{2}+\\mathrm{b}^{2}=1<\/span><\/span><\/p>\n

It tells that every point on the unit circle is defined as,<\/span><\/p>\n

a^{2}+b^{2}=1 <\/span>
\\Rightarrow \\cos ^{2} x+\\sin ^{2} x=1(\\text { Since } \\sin x=b \\text { and } \\cos x=a)<\/span>
<\/span><\/p>\n

Remember that one complete revolution subtends an angle of 2\u03c0 radian at the centre of the circle, and from the unit circle –\u00a0<\/span><\/p>\n

\\angle \\mathrm{AOB}=\\pi \/ 2, <\/span>
\\angle \\mathrm{AOC}=\\pi, <\/span>
\\angle \\mathrm{AOD}=3 \\pi \/ 2 .<\/span><\/span><\/p>\n

The coordinates of the points A, B, C and D will be (1, 0), (0, 1), (\u20131, 0) and (0, \u20131), respectively.\u00a0<\/span><\/p>\n

Now for the \u2220AOB = \u03c0\/2, we will take the coordinates of point B, which is (0, 1).<\/span><\/p>\n

This means, for the \u2220AOB = \u03c0\/2, a = 0 and b = 1<\/span><\/p>\n

Since cos x = a,\u00a0<\/span><\/p>\n

So, cos \u03c0\/2 = cos 90\u00b0 = 0<\/span><\/p>\n

Therefore, the value of the cosine of angle 90 degrees is 0.<\/span><\/p>\n

<\/span><\/p>\n

Trigonometric Method<\/b><\/h2>\n

We can derive the value of cosine of 90 degrees using sin 90 value using the below formula:<\/span><\/p>\n

\\cos ^{2} x=1-\\sin ^{2} x<\/span><\/span><\/p>\n

Putting x = 90<\/span>\u00b0<\/span><\/p>\n

\\cos ^{2} 90^{\\circ}=1-\\sin ^{2} 90^{\\circ}<\/span><\/span><\/p>\n

The value of sin 90 is 1.<\/span><\/p>\n

\\text { So, } \\cos ^{2} 90^{\\circ}=1-(1)^{2}<\/span><\/span><\/p>\n

\\cos ^{2} 90^{\\circ}=1-1 <\/span>
\\cos ^{2} 90^{\\circ}=0 <\/span>
\\cos \\left(90^{\\circ}\\right)=0<\/span><\/span><\/span><\/p>\n

 <\/p>\n

Examples<\/b><\/h2>\n

<\/b><\/p>\n

1. Evaluate: cos 90\u00b0 + sin 0\u00b0
<\/b><\/p>\n

We know that cos (90\u00b0) = sin (0\u00b0) = 0
So, cos (90\u00b0) + sin (0\u00b0)
= 0 + 0
= 0
<\/b><\/p>\n

 <\/p>\n

2. Evaluate: 2cos 90\u00b0 + 3sin 90\u00b0<\/strong><\/p>\n

We know that cos (90\u00b0) = sin (0\u00b0) = 0
and sin (90\u00b0) = cos (0\u00b0) = 1
So, 2cos (90\u00b0) + 3sin (90\u00b0)
= (2 x 0) + (3 x 1)
= 0 + 3
= 3<\/strong>\u00a0<\/p>\n

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Explore Other Topics<\/h1>\n

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\n
Geometry<\/a><\/div>\n
Trigonometry<\/a><\/div>\n
Operations<\/a><\/div>\n
Numbers<\/a><\/div>\n<\/div>\n

[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

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\n
Trigonometry Table and Trigonometry Ratios<\/a><\/div>\n
Sin 90\u00b0<\/a><\/div>\n
Sin Cos Tan Table<\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1.\u00a0<\/strong>How can you evaluate the value of the cosine of angle 90\u00b0?<\/b><\/h3>\n

    Ans:\u00a0<\/strong><\/span>We can use a unit circle that has its centre at the origin of the coordinate axes. By considering a point on this circle, we construct a right-angled triangle. By using the Pythagoras theorem and simplifying it, we can see that all the triangle angles are the integral multiples of \u03c0\/2. By taking the coordinate of angle \u03c0\/2, we can prove that cos (90\u00b0) = 0.<\/p>\n

    Q2.\u00a0<\/strong>The value of cos 90\u00b0 is equal to which value of sin?<\/b>
    <\/strong><\/h3>\n

    Ans:\u00a0<\/strong>To find the sine values in the trigonometry table, use the opposite order of the cosine function values. It means cos (90\u00b0) = sin 0\u00b0<\/span><\/p>\n

    Q3. <\/span>How will we write cos 90\u00b0 in radians?<\/b><\/h3>\n

    Ans:\u00a0<\/strong><\/span>180\u00b0 = \u03c0\u00a0 radians<\/p>\n

    So, 180\u00b0\/2 = 90\u00b0 = \u03c0\/2 radians<\/p>\n

    Therefore, cos (90\u00b0) = cos \u03c0\/2 radians.<\/p>\n

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