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Basic Applied Mathematics 2

Trigonometric Ratios

takriban dakika 4 kusoma

Mada za sehemu hiiTrigonometryMada 6
  1. Trigonometric Ratios
  2. Special Angles
  3. Radians
  4. Trigonometric identities
  5. Double angle formula
  6. Trigonometric functions

Introduction

Trigonometry is the study of the relationship between the sides and angles of a triangle. Trigonometry is applied in various real-life situations, such as solving problems related to astronomy, navigation, and building construction. Trigonometry has great practical importance to builders, architects, surveyors, engineers, and many other fields. In this chapter, you will learn how to derive and apply trigonometric ratios and identities, derive and use sine and cosine rules, evaluate trigonometric ratios using computer packages, draw graphs, and perform calculus (differentiation and integration) of trigonometric functions.

Consider ∆ORS in Figure 8.1, from which the sine, cosine, and tangent ratios can be deduced.

Right triangle O-R-S with angle theta

Figure 8.1: Right triangle O-R-S

From Figure 8.1:

  • SR is the opposite side to angle θ, with length yy units.
  • OS is the adjacent side to angle angle θ, with length xx units.
  • OR is the hypotenuse, with length rr units.

sin⁡θ=yr\sin \theta = \frac{y}{r}

cos⁡θ=xr\cos \theta = \frac{x}{r}

tan⁡θ=yx\tan \theta = \frac{y}{x}

sin⁡θ\sin \theta, cos⁡θ\cos \theta, and tan⁡θ\tan \theta are abbreviations for the sine of angle θ, cosine of angle θ, and tangent of angle θ, respectively.

Reciprocals of sine, cosine, and tangent

csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

csc⁡θ\csc \theta, sec⁡θ\sec \theta, and cot⁡θ\cot \theta are abbreviations for cosecant of angle θ, secant of angle θ, and cotangent of angle θ, respectively.

The relationship between sin θ, cos θ, and tan θ

Using ∆ORS in Figure 8.1, sin⁡θ=yr\sin \theta = \frac{y}{r}, cos⁡θ=xr\cos \theta = \frac{x}{r}, and tan⁡θ=yx\tan \theta = \frac{y}{x}. Then,

tan⁡θ=y/rx/r=yr×rx=yx=sin⁡θcos⁡θ\tan \theta = \frac{y/r}{x/r} = \frac{y}{r} \times \frac{r}{x} = \frac{y}{x} = \frac{\sin \theta}{\cos \theta}

Therefore, tan⁡θ\tan \theta can be defined in terms of sin⁡θ\sin \theta and cos⁡θ\cos \theta as:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Similarly,

cot⁡θ=x/ry/r=xy=1tan⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{x/r}{y/r} = \frac{x}{y} = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}

Rotated triangle inside the circle

Consider ∆OPQ in Figure 8.2.

Unit circle with triangle OPQ

Figure 8.2: Unit circle with triangle OPQ

The trigonometric ratios can be positive or negative depending on the quadrant in which OP lies. The values of the ratios are related to those of the corresponding acute angles.

Quadrant I

sin⁡θ (Positive)=yr\sin \theta \text{ (Positive)} = \frac{y}{r}

cos⁡θ (Positive)=xr\cos \theta \text{ (Positive)} = \frac{x}{r}

tan⁡θ (Positive)=yx\tan \theta \text{ (Positive)} = \frac{y}{x}

All trigonometric ratios in the first quadrant are positive.

Quadrant II

sin⁡(180°−θ) (Positive)=yr\sin(180° - \theta) \text{ (Positive)} = \frac{y}{r}

cos⁡(180°−θ) (Negative)=−xr\cos(180° - \theta) \text{ (Negative)} = -\frac{x}{r}

tan⁡(180°−θ) (Negative)=−yx\tan(180° - \theta) \text{ (Negative)} = -\frac{y}{x}

In the second quadrant, sine is positive, while cosine and tangent are negative.

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