Sin 90 Degrees

Sin 90 degrees

Trigonometry is the study of the affiliation between measurements of the angles of a right-angle triangle to the length of the sides of a triangle. Trigonometry is widely used by the builders to measure the height and distance of the building from its viewpoint. It is also used by the students to solve the questions based on trigonometry. The most widely used trigonometry ratios are sine, cosine, and tangent. The angels of a right angle triangle are calculated through primary functions such as sin, cosine, and tan. Other functions such as cosec, cot and secant are derived from the primary functions. Here we will study the value of sin 90 degrees and how different values will derive along with other degrees.

Sin 90 Value

\[Sin90{\text{ }}value = 1\]


As we know there are various degrees associated with the different trigonometric functions. The degrees which are widely used are O°, 30°,45°,90°,60°,180°, and 360°. We will define sin 90 degree through the below right angle triangle ABC  and with the use of both adjacent and opposite sides of a triangle and the angle of interest.

The three sides of a triangle are:

The opposite side is also known as perpendicular  and lies opposite to the angle of interest.

Adjacent Side- The point where both opposite sides and hypotenuse meet in the right angle triangle is known as the adjacent side.

Hypotenuse=-Longest side of a right-angle triangle.

As our angle of interest is Sin 90. So accordingly, the Sin function of an angle or Sin 90 degrees will be equal to the ratio of the length of the opposite side to the length of the hypotenuse side.

Sin 90 Formula

\[Sin90{\text{ }}Value{\text{ }} = \frac{{Opposite{\text{ }}side}}{{Hypotenuse{\text{ }}side}}.\]

Method to derive Sin 90 deg value

Let us calculate the Sin 90 deg value through the unit circle. The circle drawn below has radius 1 unit and the center of the circle is a place in origin.

As we know Sine function is equal to the ratio of the length of the opposite side or perpendicular to the length of the hypotenuse and considering the measurement of the adjacent side of x unit and perpendicular of 'y' unit in a right-angle triangle. We can derive Sinϴ value through our trigonometry knowledge and the figure given above.

Hence, \[\sin \theta  = \frac{1}{y}\]

Now we will measure the angle from the first quadrant to the point it reaches to the positive ‘y’ axis i.e. up to the 90°.

Now the value of y will be considered 1 as it is touching the circumference of the circle. Therefore we can say the value of y equals to 1.

\[\sin \theta  = \frac{1}{y}or\frac{1}{1}\]

Hence, Sin 90 deg will be equal to its fractional value i.e. 1/1.

Sin 90 value =1

The most widely used Sin functions in trigonometry are:-

\[\sin \left( {{{90}^o} + \theta } \right) = \cos \theta \]

\[\sin \left( {{{90}^o} - \theta } \right) = \cos \theta \]

Few other Sine identities used in trigonometry are:

\[{\text{sinx}} = \frac{1}{{\cos x}}\]

\[{\sin ^2} + {\cos ^2}x = 1\]

\[\sin \left( { - x} \right) =  - \sin x\]

\[\sin 2x = 2\sin x\cos x\]

Similarly, we can derive other values of Sin degree such as O°, 30°,45°,90°,60°,180°, and 360°.

Here in the below table, you can find out the Sine values of different angles along with various other trigonometry ratios.

Trigonometry Ratios Value

Angles in Degrees

0

30

45

60

90

Sin

0

\[\frac{1}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{{\sqrt 3 }}{2}\]

1

Cos

1

\[\frac{{\sqrt 3 }}{2}\]

\[\frac{1}{{\sqrt 2 }}\]

\[\frac{1}{2}\]

0

Tan

0

\[\frac{1}{{\sqrt 3 }}\]

1

\[\sqrt 3 \]

Not defined

Cosec

Not defined

2

\[\sqrt 2 \]

\[\frac{2}{{\sqrt 3 }}\]

1

Sec

1

\[\frac{2}{{\sqrt 3 }}\]

\[\sqrt 2 \]

2

Not defined

Cot

Not defined

\[\sqrt 3 \]

1

\[\frac{1}{{\sqrt 3 }}\]

0

 

Solved Examples

  1. Find the value of Sin 150°

Solution:

\[Sin{\text{ }}150^\circ  = {\text{ }}Sin{\text{ }}\left( {90^\circ  + 60^\circ } \right)\]

\[ = Cos60^\circ \left\{ {Since,\left( {90 + \theta } \right) = Cos\theta } \right\}\]

\[\frac{1}{2}\]

  1. Find the value of

\[{\mathbf{Tan}}\left( {{\mathbf{45}}^\circ } \right) + \left( {{\mathbf{Cos}}{\text{ }}{\mathbf{0}}^\circ } \right) + {\mathbf{Sin}}\left( {{\mathbf{90}}^\circ } \right) + {\mathbf{Cos}}\left( {{\mathbf{60}}} \right)^\circ \]

Solution:

As we know,

Tan (45°) = 1

Sin (90°) =1

Cos (0°) =1

 \[COS\left( {{{60}^O}} \right) = \frac{1}{2}\]

Now substituting the values:-

\[ = 1 + 1 + 1 + \frac{1}{2}\]

\[ = 3 + \frac{1}{2}\]

= 3.5

Fun Facts

  • Sin inverse is denoted as Sin-1 and it can also be written as arcsin or asine

  • Hipparchus is known as the Father of Trigonometry. He also discovered the values of arc and chord for a series of angles.

Quiz Time

1. If x and y are considered as a complementary angle, then

a.       Sin x=Sin y

b.      Tan x= Tan y

c.       Cos x= Cos y

d.      Sec x= Cosec y

 2.   What will be the minimum value of Sin A, 0< A <90°

a.   -1

b.  0

c.   1

d.  \[\frac{1}{2}\]