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

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 Value =\[\frac{\textrm{opposite side}}{\textrm{hypotenuse side}}\]

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, 

sinθ = 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.

Sinθ = 1/y or 1/1

Hence, Sin 90° 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(90°+θ) = cosθ

sin(90°−θ) = cosθ

Few Other Sine Identities used in Trigonometry are:

\[sinx=\frac{1}{cosx}\]

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

\[sin(-x)=-sinx\] 

\[sin2x = 2sinx cosx\]

Similarly, we can derive other values of Sin degree such as 0°, 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

Solved Examples

1.Find the value of Sin 150°

Solution:

Sin 150°= Sin (90°+60°)

=Cos 60°{Sin(90+θ)=Cosθ}

=1/2

2. Find the value of

Tan(45°)+(Cos 0°)+Sin(90°)+Cos(60°)

Solution:

As we know,

Tan (45°) = 1

Sin (90°) =1

Cos (0°) =1

Cos (60°) = 12Cos (60°) = 12

Now substituting the values:-

=1+1+1+12

=3+12

= 15

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

1. Sin x = Sin y

2. Tan x = Tan y

3. Cos x = Cos y

4. Sec x = Cosec y

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

1. -1

2. 0

3. 1

4. ½

Answers

1. Sec x = Cosec y

2. ½

More About 90 Degrees 

The trigonometric functions connect a triangle's angles to its side lengths. Trigonometric functions are useful in the study of periodic phenomena such as sound and light waves, as well as a variety of other fields. The sine, cosine, and tangent functions are the three most common trigonometric ratios. Trigonometric functions are generally defined as the ratio of two sides of a right triangle containing the angle for angles less than a right angle, and their values can be the length of various line segments around a unit circle.

The angles are determined using the primary functions of sin, cos, and tan, while the secondary functions of cosecant, secant, and cot are obtained from the primary functions. 0°, 30°, 45°, 60°, 90°, 180°, 270°, and 360° are the most common degrees. You'll learn how to compute the value of sin 90 degrees, as well as other degrees and radian quantities.

90 - Degree Sine Value

Begin by creating a right-angled triangle ABC with the angle of interest and the triangle's sides to construct the sine function of an acute angle. The following are the three sides of the triangle:

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• The opposing side is the side that is perpendicular to the angle of interest.

• The hypotenuse side is the longest side of a right triangle and is the opposite side of the right angle.

• The adjacent side is the triangle's remaining side, and it is a side of both the angle of interest and the right angle

The sine function of an angle is equal to the opposing side's length divided by the hypotenuse side's length, and the formula is as follows:

\[sin\theta =\frac{\textrm{opposite side}}{\textrm{hypotenuse side}}\]

The sides of a triangle are proportional to the sine of the opposite angles, according to the sine law.

\[\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}\]

The sine rule is used in the following examples. These are the circumstances.

Case 1: Two angles and one side are given (AAS and ASA)

Case 2: Given two sides and an angle that is not mentioned (SSA)

Finding the Value of Sin 90 Degrees via Derivation

Let's see what the value of sin 90° is. Consider the circle of the unit. That is a circle with a radius of one unit and a center at the origin.

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We may deduce from our fundamental trigonometry understanding that the base of the provided right-angled triangle measures 'x' units and the perpendicular measures 'y' units.

We are aware of this.

Sine is equal to the ratio of the length of the opposing side to the length of the hypotenuse side for any right-angled triangle measured with any of the angles. So, based on the graph,

sinθ = y/1 = 1/1

As a result, the fractional value of sin 90 degrees is 1/ 1.

90° Sin = 1

The following are the most frequent trigonometric sine functions:

theta + sin 90 degree

sin(90°+θ)=cosθ

• Sin 90 degree minus theta

sin(90°−θ)=cosθ

The following are some other trigonometric sine identities:

• \[sinx=\frac{1}{\textrm{csc x}}\]

• \[sin^2x+cos^2x=1\]

• \[sin(-x)=-sinx\]

• \[sin2x=2sinxcosx\]