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Chapter 7: Problem 18

(a) \(\sin \theta \simeq \tan \theta\). (b) \(\theta\) is very small.

Short Answer

Expert verified
\(\sin \theta \approx \tan \theta\) for very small \(\theta\).

Step by step solution

01

Understand the Approximation

For very small angles, we know that \(\theta \) in radians is approximated by \(\theta \) itself. Therefore, \(\tan \theta \approx \theta\) and \(\tan \theta \approx \sin\theta\) are well-established approximations.
02

Apply the Small Angle Approximations

Given that \(\theta\) is very small, both \(\tan \theta\) and \(\theta\) can be approximated as \(\theta\) and \(\theta\), respectively. Mathematically this can be written as: \(\tan \theta \approx \theta\) and \(\tan \theta\approx \sin\theta\).
03

Show the Equivalence

Since \(\theta\) is very small and the approximations hold, equate \(\tan \theta\) and \(\sin \theta\) to \(\theta\): \(\sin \theta \approx \tan \theta \approx \theta.\) Hence, \(\sin \theta \approx \tan \theta\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sin theta
The trigonometric function \(\text{sin} \theta\) is often encountered in the study of circles and oscillations. It helps to describe the relationship between the angle and the ratio of the length of the opposite side to the hypotenuse in a right triangle.

When \(\theta\) is very small, \(\text{sin} \theta\) becomes almost identical to \(\theta\) measured in radians. This is because, in the limit as \(\theta\) approaches zero, the arc length (which \(\text{sin} \theta\) represents) is almost the same as the straight-line distance.\
tan theta
In trigonometry, \(\text{tan} \theta\) refers to the ratio of the length of the opposite side to the adjacent side in a right triangle. For small angles, similar to \(\text{sin} \theta\), \(\text{tan} \theta\) also simplifies.

For very small values of \(\theta\), the mathematical approximation \(\text{tan} \theta \approx \theta\) holds true. This is because the difference between the arc and the tangent line becomes negligible.

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Most popular questions from this chapter

(a) Express \(7 \sin x-24 \cos x\) in the form \(R \sin (x-\alpha)\), where \(R\) is positive and \(\alpha\) is an acute angle. Hence or otherwise solve the equation $$ 7 \sin x-24 \cos x=15, \quad \text { for } 0^{\circ}

If \(x=1-\tan \theta\) and \(y=\sec \theta\) the Cartesian equation given by eliminating \(\theta\) is: (a) \(x^{2}+y^{2}=2 x\) (b) \(x^{2}-y^{2}=2 x\) (c) \(x^{2}-y^{2}+2=2 x\) (d) \((1-x)^{2}=(y-1)(y+1)\) (e) \((x-1)^{2}=(1-y)(1+y)\).

(a) \(\cos \mathrm{A}=\sin \mathrm{B}\). (b) \(\mathrm{A}+\mathrm{B}=90^{\circ}\)

(a) Prove that \(\sin ^{2} 2 \theta\left(\cot ^{2} \theta-\tan ^{2} \theta\right)=4 \cos 2 \theta\). (b) Solve the equation \(\sec \theta \tan \theta=2\), giving solutions for \(0^{\circ} \leqslant \theta<360^{\circ} .\)

(a) Find, in radians, the general solution of the equation \(4 \sin \theta=\sec \theta\). (b) If \(\sin \theta+\sin 2 \theta+\sin 3 \theta+\sin 4 \theta=0\), show that \(\theta\) is either a multiple of \(\frac{1}{2} \pi\) or a multiple of \(\frac{2}{5} \pi\).
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