Trigonometric Ratios of Some Specific Angles

# Trigonometric Ratios of Some Specific Angles | Mathematics for SSS 1 PDF Download

Trigonometry is all about triangles or to more precise about the relation between the angles and sides of a right-angled triangle. In this article we will be discussing about the ratio of sides of a right-angled triangle respect to its acute angle called trigonometric ratios of the angle and find the trigonometric ratios of specific angles: 0°, 30°, 45°, 60°, and 90°.

Consider the following triangle:

The side BA is opposite to angle ∠BCA so we call BA the opposite side to ∠C and AC is the hypotenuse, the other side BC is the adjacent side to ∠C.

### Trigonometric Ratios of angle C

Sine: Sine of ∠C is the ratio between BA and AC that is the ratio between the side opposite to C and the hypotenuse.
Sin C = BA/AC

Cosine: Cosine of ∠C is the ratio between BC and AC that is the ratio between the side adjacent to C and the hypotenuse.
Cos C = BC/AC

Tangent: Tangent of ∠C is the ratio between BA and BC that is the ratio between the side opposite and adjacent to C
Tan C = BA/BC

Cosecant: Cosecant of ∠C is the reciprocal of sin C that is the ratio between the hypotenuse and the side opposite to C.
csc C = BA/AC

Secant: Secant of ∠C is the reciprocal of cos C that is the ratio between the hypotenuse and the side adjacent to C.
sec C = BA/AC

Cotangent: Cotangent of ∠C is the reciprocal of tan C that is the ratio between the side adjacent to C and side opposite to C.
Cot C = BA/AC

### Finding trigonometric ratios for angle 0°, 30°, 45°, 60°, 90°

A. For angles 0° and 90°
If an angle A = 0° then the length of the opposite side would be zero and hypotenuse = adjacent side and if A = 90° then the hypotenuse = opposite side. So by using the above formulas for the trigonometric ratios and if the length of the hypotenuse is a.
if A = 0°

if A = 90°

Here csc 0, cot 0, tan 90 and sec 90 are not defined as at the particular angle it is divided by 0 which is undefined.

B. For angles 30° and 60°

Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore, ∠A = ∠B = ∠C = 60°.
∆ABD is a right triangle, right-angled at D with ∠BAD = 30° and ∠ABD = 60°, Here ∆ADB and ∆ADC are similar as they are Corresponding parts of Congruent triangles(CPCT).

Now we know the values of AB, BD, and AD, So the trigonometric ratios for angle 30 are

For angle 60°

C. For angle 45°
In a right-angled triangle if one angle is 45° then the other angle is also 45° thus making it an isosceles right angle triangle
If the length of side BC = a then length of AB = a and Length of AC(hypotenuse) is a√2, then

Result

The document Trigonometric Ratios of Some Specific Angles | Mathematics for SSS 1 is a part of the SSS 1 Course Mathematics for SSS 1.
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## Mathematics for SSS 1

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## FAQs on Trigonometric Ratios of Some Specific Angles - Mathematics for SSS 1

 1. What are the trigonometric ratios of the angles 0°, 30°, 45°, 60°, and 90°?
Ans. The trigonometric ratios of the angles are as follows: - For 0°: sin(0°) = 0, cos(0°) = 1, tan(0°) = 0 - For 30°: sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3 - For 45°: sin(45°) = √2/2, cos(45°) = √2/2, tan(45°) = 1 - For 60°: sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3 - For 90°: sin(90°) = 1, cos(90°) = 0, tan(90°) = undefined
 2. How can trigonometric ratios be used to solve problems involving angles in a triangle?
Ans. Trigonometric ratios can be used to solve problems involving angles in a triangle by applying the properties of sine, cosine, and tangent. These ratios relate the side lengths of a triangle to its angles. By knowing the values of any two sides or angles, we can use the appropriate trigonometric ratio to find the unknown side or angle. This is particularly useful in solving problems related to height and distance, navigation, engineering, and physics.
 3. What is the relationship between trigonometric ratios and the unit circle?
Ans. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. The trigonometric ratios (sine, cosine, and tangent) of an angle are directly related to the coordinates of the point where the terminal side of the angle intersects the unit circle. For example, the sine of an angle is equal to the y-coordinate of the point, the cosine is equal to the x-coordinate, and the tangent is equal to the ratio of the y-coordinate to the x-coordinate.
 4. How can I use the trigonometric ratios to find missing side lengths in right-angled triangles?
Ans. In a right-angled triangle, the trigonometric ratios can be used to find missing side lengths. For example, if we know the length of one side and the measure of one acute angle, we can use the sine, cosine, or tangent ratio to find the length of another side. If we know the lengths of two sides, we can use the inverse trigonometric functions (arcsine, arccosine, or arctangent) to find the measure of an angle. By applying these ratios and functions correctly, we can determine the unknown side lengths in right-angled triangles.
 5. Can trigonometric ratios be used for angles greater than 90°?
Ans. Trigonometric ratios can be used for angles greater than 90°, but their values may vary depending on the quadrant in which the angle lies. In the first quadrant (0° to 90°), the ratios are positive. In the second quadrant (90° to 180°), only the sine ratio is positive. In the third quadrant (180° to 270°), only the tangent ratio is positive. In the fourth quadrant (270° to 360°), only the cosine ratio is positive. Beyond 360°, the trigonometric ratios repeat their values. However, it's important to note that angles greater than 90° are not commonly used in basic trigonometry and are usually considered in advanced topics.

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