How to Calculate With Angles

Calculations with angles are an important part of trigonometric mathematics, and remain important through more advanced mathematics disciplines, such as calculus. The three primary functions used in association with angles are sine, cosine and tangent, represented as sin, cos and tan on most calculators. These functions are used with right triangles, and the practice there can then be applied to calculations with circles.

Things You'll Need

• Protractor
• Scientific calculator

Instructions

1. Measure the angle if you do not know its degree. In most instances in which you are being tested on the principles, the angle's measure will be provided and you will then have to calculate with the given information; however, a protractor can be used to measure an angle that is not provided. Place the center of the protractor at the corner being measured, with the horizontal line along the bottom of the protractor running directly over one line of the angle, then read where the second line of the angle intersects the protractor's scale.

2. Identify which sides of the triangle you know the length of, in relation to the angle being calculated with. The opposite side is the side of the triangle that is not a part of the angle. The hypotenuse is the longest side of the triangle. The adjacent side is the side of the triangle that is a part of the triangle, but not the hypotenuse. For example, if given a hypotenuse of 5, and an angle of the triangle that is 30 degrees, the shortest side of the triangle, which is not touching the 30 degree angle, is the opposite side. The line which is longer than the opposite and shorter than the hypotenuse is the adjacent. The adjacent will always be larger than the opposite if the angle is less than 45 degrees, and smaller if the angle is greater than 45 degrees.

3. Select the appropriate formula for determining the side in question. Each formula involves two of the three sides, and can be used to find the length of one side, using the given length of the other side in the formula and the degree of the angle. The sine of an angle is equal to the length of the opposite divided by the length of the hypotenuse, and can thus be used to find the hypotenuse when given the opposite, or find the opposite when given the hypotenuse. The cosine of an angle is equal to the length of the adjacent divided by the length of the hypotenuse. The tangent of an angle is equal to the length of the opposite over the length of the adjacent. In the previous example, the sine function would be used if asked to determine the length of the shortest side of the triangle.

4. Calculate the value of the angle side of the equation, by finding the sine, cosine or tangent of the angle, as determined in Step 2. With a graphing calculator, press the corresponding button (sin, cos or tan) and then enter the angle into the parenthetical brackets. With a simpler calculator, such as Windows' basic calculator, enter the angle into the calculator, then press the corresponding button. In the above example, sin(30) is entered into the calculator (or, 30 is entered then "sin" is pressed on a basic calculator) leading to a result of 1/2.

5. Calculate for the side of the triangle that is unknown. If the unknown side is on the bottom, and thus being divided by, this is accomplished by dividing the known side by the result of Step 4. If the unknown side is on top, and thus the number being divided, this is accomplished by multiplying the known side by the result of Step 4. As the hypotenuse is known in the above example, and sine of a degree is equal to the opposite over the hypotenuse, our result in Step 4 must be multiplied by the hypotenuse. Multiply 5 by 1/2, to find that the length of the shortest side is 2 1/2.

Tips & Warnings

• SoCaToa is a mnemonic device to remember the formalas for angle calculation. SoCa represents which side is placed over the hypotenuse, Sine is opposite (So) and Cosine is adjacent (Ca). Toa represents that Opposite is placed over Adjacent with a tangent.

References

• Easy Calculations: Angles and Sides Calculations