Trigonometric Ratios Of Standard Angles - GMS - Learning Simply
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Trigonometric Ratios Of Standard Angles

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Trigonometric Ratios Of Standard Angles

Trigonometric ratios are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant. The standard angles for these trigonometric ratios are 0°, 30°, 45°, 60° and 90°. These angles can also be represented in the form of radians such as 0, π/6, π/4, π/3, and π/2. These angles are most commonly and frequently used in trigonometry. Learning the values of these trigonometry angles is very necessary to solve various problems. 




Trigonometric Ratios Formulas:

The six trigonometric ratios are basically expressed in terms of the right-angled triangle.

Trigonometric Ratios

∆ABC is a right-angled triangle, right-angled at (shown in figure 1). The six trigonometric ratios for ∠C are defined as:

sin C = ABAC

cosec C = 1sin C

cos C = BCAC

sec C = 1cos C

tan C = sin Ccos C                                

cot C = 1tan C

The standard angles for which trigonometric ratios can be easily determined are 0°,30°,45°,60° and 90°. The values are determined using properties of triangles. The two acute angles of a right-angled triangle are complementary.

Trigonometric Ratios Table (Standard Angles)

Angle = ∠C30°45°60°90°
sin C01212321
cos C13212120
tan C01313Not Defined
cosec CNot Defined22231
sec C12322Not Defined
cot CNot Defined31130

The above table shows the important angles for all the six trigonometric ratios. Let us learn here how to derive these values.

Derivation of Trigonometric Ratios for Standard Angles

Value of Trigonometric Ratios for Angle equal to 45 degrees

Trigonometric Ratios

In ABC, if C = 45°, then A = 45°. Since the angles are equal, ABC becomes a right angled isosceles triangle. So, AB = BC. Assume AB = BC = a units.

Using Pythagoras theorem ,

AC2 = AB2 + BC2

AC2 = a2 + a2

AC = a2 units

C = 45°

 sin C = sin 45° = ABAC = aa2 = 12                                 

cosec 45° = 1sin 45° = 2

cos C = cos 45° = BCAC = aa2 = 12                                         

sec 45° = 1cos 45° = 2

tan 45° = sin 45°cos 45° = a2a2 = 1                                                           

cot 45° = 1tan 45° = 1

Value of Trigonometric Ratios for Angle equal to 30 and 60 degrees

Trigonometric Ratios

In figure 3, ΔPQR is equilateral. The perpendicular from any vertex on the opposite side is coincident with the angle bisector of that particular vertex. Also, the perpendicular bisects the opposite side. If a perpendicular PS is dropped on QR, then QPS = SPR = 30° and QS = SR. Assume PQ = QR = RP = 2a units.

 QS = SR = a units

In ΔPSQ, by Pythagoras theorem,

PQ2 = QS2 + PS2

PS2 = (2a)2  a2

PS = 3a2 = 3a

SPQ = 30°

sin SPQ = sin 30° = SQPQ = a2a = 12                                        

cosec 30° = 1sin 30° = 2

cos SPQ = cos 30° = PSPQ = 3a2a = 32                                    

sec 30° = 1cos 30° = 23

tan 30° = sin 30°cos 30° = 1232 = 13                                                        

cot 30° = BCAB = 3

Similarly, ratios of 60° are determined by finding the ratios of SQP as

sin 60° = 32                                                          

cos 60° = 12

tan 60° = 3