Understanding Angle Formulas and Trigonometry
Comprehensive Definition, Description, Examples & RulesÂ
A branch of mathematics that helps us find the relationship between the angles and sides of triangles is called trigonometry. As an important mathematical operation, trigonometry plays an integral part in miscellaneous domains like engineering, physics, navigation, etc To get to know trigonometry fully, one needs to understand varied elemental ideas like sides, angles, and unit circles, along with important trigonometric functions like sine, cosine, tangent, etc:
Angles
An angle is just a dimension of the amount of rotation between two rays that share a common endpoint, called the vertex. Angles are mostly measured in degrees (°) or radians (rad). Radians are more used in angle formulas in trigonometry because they are more convenient when performing mathematical operations. A full circle is analogous to 360 degrees or 2π radians.
Sides of a Right Triangle
Trigonometry uses right triangles with one angle equal to 90 degrees. In a right-angle triangle, there are three basic sides:
- Â Â Â Â Hypotenuse: The side that lies opposite to the right angle is known as the hypotenuse and is the longest side in a right-angle triangle.
- Â Â Â Â Adjacent Side: The side of the triangle that is adjacent to the angle being considered is known as the adjacent side.
- Â Â Â Â Opposite Side: The side of the triangle opposite to the considered angle is known as the adjacent side.
Unit Circle
A circle with a radius of 1 unit centred at the origin of a coordinate system is known as the unit circle. Every point on the circle refers to an angle, and the coordinates of that point represent values of the sine and cosine of the angle.
For an angle θ in a position where the first side of the angle starts along the positive x-axis, the coordinates of the point lying on the unit circle corresponding to θ are (cos (θ), sin (θ)).
Trigonometric Functions
Trigonometric functions are mathematical operations that connect a right triangle’s angles to its sides’ proportions. The three standard trigonometric functions are sine, cosine and tangent:
- Â Â Â Â Sine (SIN): The sine of an angle in a right triangle is the ratio of the length of the side opposite of that angle to the length of the hypotenuse.
SIN θ = opposite/ hypotenuse
- Â Â Â Â Cosine (COS): The cosine of an angle in a right triangle is the ratio of the length of the side adjacent to that angle to the length of the hypotenuse.
COS θ = adjacent/ hypotenuse
- Â Â Â Â Tangent (TAN): The tangent of a right triangle is the ratio of the length of the side opposite of that angle to the length of the side adjacent to that angle.
TAN θ = opposite/adjacent
The Cosine Rule and Calculating Angles
The cosine rule to find angles is an important trigonometric formula used to find measures of angles in triangles when you know the lengths of two sides and the included angle. It comes in handy when trying to calculate angles in non-right triangles.
Formula for Cosine Rule
In a triangle with sides of lengths a, ‘b’, ‘c’, and an angle ‘C’ opposite side ‘c’, the cosine rule stands as follows:
 c^2 = a^2 + b^2 – 2ab*cos(C)
Here’s how you can use the cosine rule to find an angle when you have two sides and the included angle:
- Identify the lengths of two sides of the triangle (a and b) and the length of the included angle (C).
- Employ these numerals in the formula above.Â
- You can rearrange the formula equation to find the included angle (C).
Calculating Angles from Two Sides
To know the application of the cosine rule and understand how to find angles when only two sides of a triangle are known, let’s go through the following example:
Find the measure of angle A in a triangle with side a= 8 units, b= 10 units and the measure of the included angle C as 45 degrees.
Solution:
1)Â Â Put the given values in the cosine rule formula;
c^2 = a^2 + b^2 – 2ab*cos(C)
Here,
A= 8
B= 10
C= 45
2)Â Â Rearrange to solve for A
A= COS^-1 (a^2+b^2-c^2/2ab)
Hence, COS A= a^2+b^2-c^2/2ab
3)Â Â Substitute the given values
So, COS A= 8^2+10^2-c^2/2*8*10
COS A= 64+100-C^2/160
COS A= 164-C^2/160
COS A= 0.707
The degree corresponding to the value COS A= 0.707 is 45 degrees. So, the measure of angle A is approximately 45 degrees.
Finding Angles of a Triangle Using Trigonometry
Trigonometry is a very useful mathematical operation that helps us find the measures of angles when we have the information on the lengths of the sides by applying trigonometric functions like sine, cosine, and tangent along with inverse trigonometric functions. To understand how to find the angles of a triangle, two alternative methods used for calculating angles in such cases are the sine rule and tangent rule:
Sine Rule
The sine rule is used when you know the lengths of two sides of a right triangle and the measure of an angle that does not fall between the given sides. It relates to the ratio of the lengths of the sides to the sines of their opposite angles. The sine rule is mathematically represented as:
a/SIN (A)= b/SIN (B)= c/SIN (C)
Where a, b, and c represent the dimension of sides.
A, B, and C are measurements of the angles opposite the individual sides. Â
Tangent Rule
If you know the measurements of two sides and the value of the included angle, i.e., the angle that falls between the given sides, the tangent rule can be used as the tangent of half the difference of the angles formed by the given sides corresponds to the tangent of half the sum of those angles. The tangent rule is represented as:
Tan (A-B/2)/Tan (A+B/2)= a-b/a+b= c-d/c+d
Where A and B are used to denote angles formed by the given sides
A and b denote the measurement of the lengths of the sides.Â
C and d represent the measurement of the lengths of sides.
Methodical Approach to Finding Angles
If you want to know the process of finding angles of a triangle, here is a systematic approach to solving angle-related problems using trigonometric rules:
1)Â Â Look for side lengths and angle measures and check whether you have all three sides, two sides and an included angle, or two sides and a non-included angle.
2)Â Â Determine what you need to compute.
3) Select the appropriate method based on the given information:
- Use the cosine rule to find one of the angles if you have the measurement of all three sides.Â
- Use the sine rule to find one of the other angles if you have the measurement of two sides and an included angle.
- Use the tangent rule to find one of the other angles if you have the measurement of two sides and a non-included angle.
Real-Life Applications
Trigonometry is important in a variety of real-world scenarios, such as:
Architects can develop safer structures for us that are also solid and durable only through the use of angles in trigonometry.
Navigators get to know about their precise location and directions using trigonometric functions and computing angles.Â
Research and survey about land is possible using trigonometric formulas to calculate entities like heights, lengths and angles.Â
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Key Takeaways
- Trigonometry helps us determine the relationship between the angles and sides of triangles.
- The three rudimentary trigonometric functions are sine, oscine, and tangent.
- Trigonometry is used in a range of real-world scenarios like navigation, architecture and conducting surveys.
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Frequently Asked Questions
You can find the missing angle in a triangle by making use of the known fact that the sum of all angles in a triangle is 180 degrees.Â
To find angles in a quadrilateral, you need to minus the known angles from 360 degrees in the case of a convex quadrilateral and 180 degrees in the case of a concave quadrilateral.Â
To calculate the angles in a circle or a sector, you should follow the formula: Angle= (Aec length/ circle circumference)* 360.
To calculate angles in a right triangle using trigonometry, you can make use of sine, cosine and tangent functions according to the provided information.Â
The formula for calculating the exterior angles of a polygon is Exterior Angle = 360 degrees / Number of sides.
