Question

Given the approximation \(\sin 38^{\circ}=0.62,\) use trigonometric identities to find the approximate value of (a) \(\cos 38^{\circ}\) (b) \(\tan 38^{\circ}\) (c) \(\cot 38^{\circ}\) (d) \(\sec 38^{\circ}\) (e) \(\csc 38^{\circ}\) (f) \(\sin 52^{\circ}\) (g) \(\cos 52^{\circ}\) (h) \(\tan 52^{\circ}\)

Short answer

Approximations: \(\text {cos} \ 38^{\text {o}} \approx 0.7846\), \(\text {tan} \ 38^{\text {o}} \approx 0.79\), \(\text {cot} \ 38^{\text {o}} \approx 1.27\), \(\text {sec} \ 38^{\text {o}} \approx 1.274\), \(\text {csc} \ 38^{\text {o}} \approx 1.613\), \(\text {sin} \ 52^{\text {o}} \approx 0.7846\), \(\text {cos} \ 52^{\text {o}} \approx 0.62\), \(\text {tan} \ 52^{\text {o}} \approx 1.27\).

Step-by-step solution

- Identify Given Information

We are given that \(\text {sin} \ 38^{\text {o}} = 0.62\). We'll use this to find the other trigonometric values.

- Find \(\text{cos} \ 38^{\text {o}}\)

Using the equation \[\sin^2 \ x + \cos^2 \ x = 1 \], we can solve for \({cos 38^{\text {o}}}\): \[\cos^2 38^\circ = 1 - \sin^2 38^\circ \] \[\approx 1 - (0.62)^2 \] \[\approx 1 - 0.3844 = 0.6156 \] \[\cos 38^\circ \approx \sqrt{0.6156} \approx 0.7846 \]

- Find \(\text{tan} \ 38^{\text {o}}\)

Using the equation \[\tan x = \frac{\sin x}{\cos x} \], we can find \({\tan \ 38^{\text {o}}}\): \[\tan 38^\circ = \frac{\sin 38^\circ}{\cos 38^\circ} \] \[\approx \frac{0.62}{0.7846} \approx 0.79 \]

- Find \(\text{cot} \ 38^{\text {o}}\)

Using the equation \[\cot x = \frac{1}{\tan x} \], we can find \({cot \ 38^{\text {o}}}\): \[\cot 38^\circ \approx \frac{1}{0.79} \approx 1.27 \]

- Find \(\text{sec} \ 38^{\text {o}}\)

Using the equation \[\sec x = \frac{1}{\cos x} \], we can find \({sec 38^{\text {o}}}\): \[\sec 38^\circ \approx \frac{1}{0.7846} \approx 1.274 \]

- Find \(\text{csc} \ 38^{\text {o}}\)

Using the equation \[\csc x = \frac{1}{\sin x} \], we can find \({csc 38^{\text {o}}}\): \[\csc 38^\circ \approx \frac{1}{0.62} \approx 1.613 \]

- Find \(\text{sin} \ 52^{\text {o}}\)

Using the identity \[\sin(90^\circ - x) = \cos x \], we find \[\sin 52^\circ = \cos 38^\circ \approx 0.7846 \]

- Find \(\text{cos} \ 52^{\text {o}}\)

Using the identity \[\cos(90^\circ - x) = \sin x \], we find \[\cos 52^\circ = \sin 38^\circ \approx 0.62 \]

- Find \(\text{tan} \ 52^{\text {o}}\)

Using the identity \[\tan(90^\circ - x) = \cot x \], we find \[\tan 52^\circ = \cot 38^\circ \approx 1.27 \]

Key concepts

Sine and Cosine

Sine and cosine are fundamental trigonometric functions. They relate the angles of a right triangle to the lengths of its sides. The sine of an angle in a right triangle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse, expressed as: \[ \text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}} \] The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse: \[ \text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \] These functions are complementary to each other. If you have the value of sine, you can find the cosine using the Pythagorean identity: \[ \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \] For example, if \(\text {sin} 38^{\text {o}} = 0.62,\) you can find cosine as: \[ \text{cos} 38^\text{o} = \sqrt{1 - (\text{sin} 38^\text{o})^2} \approx 0.7846 \] Understanding this relationship is crucial for solving a wide range of trigonometric problems.

Tangent and Cotangent

Tangent and cotangent are also essential trigonometric functions. The tangent of an angle is the ratio of the sine of the angle to the cosine of the angle: \[ \text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \] For instance, if \(\text{sin} 38^{\text{o}} = 0.62\) and \(\text{cos} 38^\text{o} = 0.7846,\) then: \[ \text{tan} 38^{\text {o}} \approx \frac{0.62}{0.7846} \approx 0.79 \] Cotangent is the reciprocal of tangent: \[ \text{cot} \theta = \frac{1}{\text{tan} \theta} \] So, \[ \text{cot} 38^\text{o} \approx \frac{1}{0.79} \approx 1.27 \] Understanding tangent and cotangent helps in solving real-world problems involving slopes and angles of elevation.

Secant and Cosecant

Secant and cosecant are reciprocals of cosine and sine, respectively. They provide more ways to relate angles and sides in right triangles. The secant of an angle is the reciprocal of the cosine of that angle: \[ \text{sec} \theta = \frac{1}{\text{cos} \theta} \] Using the example where \(\text{cos} 38^\text{o} = 0.7846,\) we get: \[ \text{sec} 38^\text{o} \approx \frac{1}{0.7846} \approx 1.274 \] The cosecant is the reciprocal of the sine: \[ \text{cosec} \theta = \frac{1}{\text{sin} \theta} \] So, \[ \text{cosec} 38^\text{o} \approx \frac{1}{0.62} \approx 1.613 \] Knowing these functions can help in computations where cosine or sine alone might not simplify enough to easily determine answers.

Trigonometric Functions

Trigonometric functions form a comprehensive system to relate angles to side lengths in right triangles. The primary functions are sine (\text{sin}), cosine (\text{cos}), and tangent (\text{tan}). Secondary functions such as cotangent (\text{cot}), secant (\text{sec}), and cosecant (\text{csc}) are derived from the primary ones. Each function provides unique insights and methods to solve problems. The relationships are defined as: - \[ \text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \] - \[ \text{cot} \theta = \frac{1}{\text{tan} \theta} \] - \[ \text{sec} \theta = \frac{1}{\text{cos} \theta} \] - \[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \] These relationships paved the way for solving triangles with non-right angles and modeling periodic phenomena like sound and light waves.

Angle Complementary Relationships

Angles are complementary if their sum is 90 degrees. Leveraging this concept can simplify solving trigonometric equations. For example, \(38^\text {o} + 52^\text {o} = 90^\text {o}\). Therefore: - \[ \text{sin}(90^\text {o} - \theta) = \text{cos} \theta \] - \[ \text{cos}(90^\text {o} - \theta) = \text{sin} \theta \] - \[ \text{tan}(90^\text {o} - \theta) = \text{cot} \theta \] From the solved examples, given \(\text{sin} 38^\text{o} = 0.62\): - \[ \text{sin} 52^\text {o} = \text{cos} 38^\text {o} = 0.7846 \] - \[ \text{cos} 52^\text {o} = \text{sin} 38^\text {o} = 0.62 \] - \[ \text{tan} 52^\text {o} = \text{cot} 38^\text {o} = 1.27 \] These identities simplify calculations and are helpful in understanding geometric properties. Sharing this understanding enables tackling more complex trigonometric problems effectively.