Without using a calculator, find the five other...

Without using a calculator, find the five other trigonometric ratios [including $\sec (\theta), \operatorname{cosec}(\theta)$ and $\cot (\theta)]$ given:
i $\sin (\theta)=\frac{1}{\sqrt{3}} \quad$ ii $\cos (\theta)=\frac{2}{3}$
iii $\tan (\theta)=\frac{3}{4}$

Key Concepts

Deriving Trigonometric Ratios from Known Values

When one trigonometric function is given, other ratios can be derived using the relationships provided by reciprocal, Pythagorean, and quotient identities. This method allows for the complete determination of all trigonometric values related to an angle, which is especially useful in problems that preclude the use of a calculator.

Quotient Identities

Quotient identities express the tangent and cotangent functions as the ratio of sine to cosine and cosine to sine, respectively. For example, tan(?) = sin(?)/cos(?) and cot(?) = cos(?)/sin(?). This relation is particularly useful when converting between different trigonometric ratios.

Pythagorean Identities

Pythagorean identities, such as sin²(?) + cos²(?) = 1, are essential tools in trigonometry that relate the squares of sine and cosine. They provide a way to compute one function when the other is known, and are often used to ensure the consistency of derived values or to resolve ambiguities in sign.

Reciprocal Identities

Reciprocal identities establish that certain trigonometric functions are the reciprocals of others; for instance, cosecant is the reciprocal of sine (cosec(?) = 1/sin(?)), secant is the reciprocal of cosine (sec(?) = 1/cos(?)), and cotangent is the reciprocal of tangent (cot(?) = 1/tan(?)). These identities allow one to derive one function from another when only a single function's value is provided.

Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—describe the relationships between angles and side lengths in right-angled triangles. They are fundamental in trigonometry and are used in various applications ranging from geometric problems to real-world phenomena modeling.

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