In \(\triangle \mathrm{CDE}, D\) is a right angle. The lengths of \(\mathrm{CD}, \mathrm{DE}\) and \(\mathrm{CE}\) are \(\alpha, \beta\) and \(\gamma\) respectively. State (a) \(\sin C\) (b) \(\cos C\) (c) \(\tan C\) (d) \(\sin E\) (e) \(\tan E(\) f) \(\cos E\)

Short Answer

Expert verified
Answer: The trigonometric functions for angles C and E in triangle CDE are: (a) sin C = α/γ (b) cos C = β/γ (c) tan C = α/β (d) sin E = β/γ (e) tan E = β/α (f) cos E = α/γ

Step by step solution

01

Preliminary information

In a right-angled triangle like CDE, with right angle at vertex D, the opposite, adjacent, and hypotenuse sides are defined with respect to an angle in the triangle. For example, for angle C, CD is opposite, DE is adjacent, and CE is the hypotenuse. Similarly, for angle E, DE is opposite, CD is adjacent, and CE is the hypotenuse.
02

Part (a) and (b) - Finding \(\sin C\) and \(\cos C\)

To find \(\sin C\), we need to use the definition of sine in a right-angled triangle: \(\sin\angle{C} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\alpha}{\gamma}\). Meanwhile, \(\cos C\) can be found via the definition of cosine: \(\cos\angle{C} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\beta}{\gamma}\).
03

Part (c) - Finding \(\tan C\)

The tangent function, \(\tan C\), can be determined as \(\tan\angle{C} = \frac{\sin\angle{C}}{\cos\angle{C}} = \frac{\frac{\alpha}{\gamma}}{\frac{\beta}{\gamma}} = \frac{\alpha}{\beta}\).
04

Part (d) - Finding \(\sin E\)

To determine \(\sin E\), we can use the definition of sine: \(\sin\angle{E} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\beta}{\gamma}\).
05

Part (e) - Finding \(\tan E\)

The tangent function, \(\tan E\), can be represented as \(\tan\angle{E} = \frac{\sin\angle{E}}{\cos\angle{E}} = \frac{\frac{\beta}{\gamma}}{\frac{\alpha}{\gamma}} = \frac{\beta}{\alpha}\).
06

Part (f) - Finding \(\cos E\)

To find \(\cos E\), we can use the definition of cosine: \(\cos\angle{E} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\alpha}{\gamma}\). To conclude, the values of the trigonometric functions for angles C and E in triangle CDE are: (a) \(\sin C = \frac{\alpha}{\gamma}\) (b) \(\cos C = \frac{\beta}{\gamma}\) (c) \(\tan C = \frac{\alpha}{\beta}\) (d) \(\sin E = \frac{\beta}{\gamma}\) (e) \(\tan E = \frac{\beta}{\alpha}\) (f) \(\cos E = \frac{\alpha}{\gamma}\)

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