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Engineering Mathematics III for Mechatronics Technology professionals requires mastery of mathematical concepts essential for analyzing and designing control systems, sensors, and actuators. Understanding angles and their functions, including hyperbolic functions, is fundamental in kinematics, robotics, and signal processing. This chapter focuses on acute, obtuse, reflex, and right angles, providing mathematical tools to solve problems involving these angles in practical mechatronic applications.
Angles are critical in describing the orientation and motion of mechanical components in mechatronics systems. Accurate calculations involving angles enable precise control of robotic arms, sensor alignment, and actuator movement. Hyperbolic functions, often used in signal analysis and system modeling, complement trigonometric functions in describing non-linear behaviors.
Acute angles are angles less than 90°. In mechatronics, these angles often appear in sensor positioning and linkage mechanisms. The basic trigonometric functions for an acute angle \(\theta\) are sine, cosine, and tangent:
$$\sin \theta = \frac{\text{Opposite side}}{\text{Hypotenuse}}$$
$$\cos \theta = \frac{\text{Adjacent side}}{\text{Hypotenuse}}$$
$$\tan \theta = \frac{\text{Opposite side}}{\text{Adjacent side}}$$
Hyperbolic functions such as \(\sinh \theta\), \(\cosh \theta\), and \(\tanh \theta\) are defined similarly but relate to hyperbolas rather than circles:
$$\sinh \theta = \frac{e^{\theta} - e^{-\theta}}{2}$$
$$\cosh \theta = \frac{e^{\theta} + e^{-\theta}}{2}$$
$$\tanh \theta = \frac{\sinh \theta}{\cosh \theta}$$
Example 1: A robotic arm is set at an acute angle of 30° to the horizontal. Calculate the sine, cosine, and tangent of this angle.
Given: \(\theta = 30^\circ\)
$$\sin 30^\circ = 0.5$$
$$\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866$$
$$\tan 30^\circ = \frac{1}{\sqrt{3}} \approx 0.577$$
Answer: \(\sin 30^\circ = 0.5\), \(\cos 30^\circ = 0.866\), \(\tan 30^\circ = 0.577\)
Example 2: Calculate \(\sinh 0.5\) and \(\cosh 0.5\) for a sensor signal modeled by a hyperbolic function.
Given: \(\theta = 0.5\)
$$\sinh 0.5 = \frac{e^{0.5} - e^{-0.5}}{2}$$
$$= \frac{1.6487 - 0.6065}{2} = \frac{1.0422}{2} = 0.5211$$
$$\cosh 0.5 = \frac{e^{0.5} + e^{-0.5}}{2}$$
$$= \frac{1.6487 + 0.6065}{2} = \frac{2.2552}{2} = 1.1276$$
Answer: \(\sinh 0.5 = 0.5211\), \(\cosh 0.5 = 1.1276\)
Example 3: Find \(\tanh 0.5\) using the values from Example 2.
Given: \(\sinh 0.5 = 0.5211\), \(\cosh 0.5 = 1.1276\)
$$\tanh 0.5 = \frac{0.5211}{1.1276} = 0.4621$$
Answer: \(\tanh 0.5 = 0.4621\)
Example 4: A sensor detects a displacement that varies as \(\sinh \theta\) where \(\theta = 1\) radian. Calculate the displacement.
Given: \(\theta = 1\)
$$\sinh 1 = \frac{e^{1} - e^{-1}}{2} = \frac{2.7183 - 0.3679}{2} = \frac{2.3504}{2} = 1.1752$$
Answer: Displacement = 1.1752 units
Example 5: Calculate \(\cosh 1.5\) for a system parameter.
Given: \(\theta = 1.5\)
$$\cosh 1.5 = \frac{e^{1.5} + e^{-1.5}}{2} = \frac{4.4817 + 0.2231}{2} = \frac{4.7048}{2} = 2.3524$$
Answer: \(\cosh 1.5 = 2.3524\)
Obtuse angles measure between 90° and 180° and often appear in the analysis of mechanical linkages and joint angles in mechatronic systems. Trigonometric functions for obtuse angles require adjustments because sine remains positive while cosine and tangent become negative depending on the quadrant. Hyperbolic functions are not defined by angle size but by real number inputs, so their use remains consistent.
The sine, cosine, and tangent of an obtuse angle \(\theta\) can be related to its supplementary acute angle \(180^\circ - \theta\):
$$\sin \theta = \sin (180^\circ - \theta)$$
$$\cos \theta = -\cos (180^\circ - \theta)$$
$$\tan \theta = -\tan (180^\circ - \theta)$$
Example 1: Calculate \(\sin 120^\circ\), \(\cos 120^\circ\), and \(\tan 120^\circ\).
Given: \(\theta = 120^\circ\)
Supplementary acute angle:
$$180^\circ - 120^\circ = 60^\circ$$
$$\sin 120^\circ = \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$$
$$\cos 120^\circ = -\cos 60^\circ = -0.5$$
$$\tan 120^\circ = -\tan 60^\circ = -\sqrt{3} \approx -1.732$$
Answer: \(\sin 120^\circ = 0.866\), \(\cos 120^\circ = -0.5\), \(\tan 120^\circ = -1.732\)
Example 2: A linkage in a mechatronic arm forms an obtuse angle of 135°. Find the sine, cosine, and tangent.
Given: \(\theta = 135^\circ\)
$$180^\circ - 135^\circ = 45^\circ$$
$$\sin 135^\circ = \sin 45^\circ = \frac{\sqrt{2}}{2} \approx 0.707$$
$$\cos 135^\circ = -\cos 45^\circ = -\frac{\sqrt{2}}{2} \approx -0.707$$
$$\tan 135^\circ = -\tan 45^\circ = -1$$
Answer: \(\sin 135^\circ = 0.707\), \(\cos 135^\circ = -0.707\), \(\tan 135^\circ = -1\)
Example 3: Calculate \(\sin 150^\circ\), \(\cos 150^\circ\), and \(\tan 150^\circ\).
Given: \(\theta = 150^\circ\)
$$180^\circ - 150^\circ = 30^\circ$$
$$\sin 150^\circ = \sin 30^\circ = 0.5$$
$$\cos 150^\circ = -\cos 30^\circ = -0.866$$
$$\tan 150^\circ = -\tan 30^\circ = -0.577$$
Answer: \(\sin 150^\circ = 0.5\), \(\cos 150^\circ = -0.866\), \(\tan 150^\circ = -0.577\)
Example 4: A sensor output varies as \(\sin \theta\) where \(\theta = 110^\circ\). Calculate the sensor output value.
Given: \(\theta = 110^\circ\)
$$180^\circ - 110^\circ = 70^\circ$$
$$\sin 110^\circ = \sin 70^\circ = 0.940$$
Answer: Sensor output = 0.940 units
Example 5: For \(\theta = 140^\circ\), calculate \(\cos \theta\) and \(\tan \theta\).
Given: \(\theta = 140^\circ\)
$$180^\circ - 140^\circ = 40^\circ$$
$$\cos 140^\circ = -\cos 40^\circ = -0.766$$
$$\tan 140^\circ = -\tan 40^\circ = -0.839$$
Answer: \(\cos 140^\circ = -0.766\), \(\tan 140^\circ = -0.839\)
Reflex angles are greater than 180° but less than 360°, occurring in complex joint rotations and sensor orientations in mechatronic systems. Calculating trigonometric functions for reflex angles involves using the reference angle \(360^\circ - \theta\) and the signs of functions depending on the quadrant. Reflex angles often describe rotations beyond a straight line, important in robotic arm articulation.
The trigonometric functions for a reflex angle \(\theta\) are:
$$\sin \theta = -\sin (360^\circ - \theta)$$
$$\cos \theta = \cos (360^\circ - \theta)$$
$$\tan \theta = -\tan (360^\circ - \theta)$$
Example 1: Calculate \(\sin 210^\circ\), \(\cos 210^\circ\), and \(\tan 210^\circ\).
Given: \(\theta = 210^\circ\)
Reference angle:
$$360^\circ - 210^\circ = 150^\circ$$
Calculate \(\sin 210^\circ\):
$$\sin 210^\circ = -\sin 150^\circ = -0.5$$
Calculate \(\cos 210^\circ\):
$$\cos 210^\circ = \cos 150^\circ = -0.866$$
Calculate \(\tan 210^\circ\):
$$\tan 210^\circ = -\tan 150^\circ = -(-0.577) = 0.577$$
Answer: \(\sin 210^\circ = -0.5\), \(\cos 210^\circ = -0.866\), \(\tan 210^\circ = 0.577\)
Example 2: Find \(\sin 270^\circ\), \(\cos 270^\circ\), and \(\tan 270^\circ\).
Given: \(\theta = 270^\circ\)
Reference angle:
$$360^\circ - 270^\circ = 90^\circ$$
$$\sin 270^\circ = -\sin 90^\circ = -1$$
$$\cos 270^\circ = \cos 90^\circ = 0$$
$$\tan 270^\circ = -\tan 90^\circ = \text{undefined}$$
Answer: \(\sin 270^\circ = -1\), \(\cos 270^\circ = 0\), \(\tan 270^\circ\) undefined
Example 3: Calculate \(\sin 300^\circ\), \(\cos 300^\circ\), and \(\tan 300^\circ\).
Given: \(\theta = 300^\circ\)
Reference angle:
$$360^\circ - 300^\circ = 60^\circ$$
$$\sin 300^\circ = -\sin 60^\circ = -0.866$$
$$\cos 300^\circ = \cos 60^\circ = 0.5$$
$$\tan 300^\circ = -\tan 60^\circ = -\sqrt{3} = -1.732$$
Answer: \(\sin 300^\circ = -0.866\), \(\cos 300^\circ = 0.5\), \(\tan 300^\circ = -1.732\)
Example 4: A sensor rotates through 330°. Find the sine and cosine of this reflex angle.
Given: \(\theta = 330^\circ\)
Reference angle:
$$360^\circ - 330^\circ = 30^\circ$$
$$\sin 330^\circ = -\sin 30^\circ = -0.5$$
$$\cos 330^\circ = \cos 30^\circ = 0.866$$
Answer: \(\sin 330^\circ = -0.5\), \(\cos 330^\circ = 0.866\)
Example 5: Calculate \(\tan 315^\circ\).
Given: \(\theta = 315^\circ\)
Reference angle:
$$360^\circ - 315^\circ = 45^\circ$$
$$\tan 315^\circ = -\tan 45^\circ = -1$$
Answer: \(\tan 315^\circ = -1\)
Right angles measure exactly 90° and are fundamental in mechatronics for defining orthogonal axes, sensor calibration, and component alignment. The trigonometric functions at 90° have specific values critical for simplifying calculations:
$$\sin 90^\circ = 1$$
$$\cos 90^\circ = 0$$
$$\tan 90^\circ = \text{undefined}$$
Hyperbolic functions do not have special values at 90° since their domain is all real numbers rather than degrees.
Example 1: Calculate \(\sin 90^\circ\), \(\cos 90^\circ\), and \(\tan 90^\circ\).
Given: \(\theta = 90^\circ\)
$$\sin 90^\circ = 1$$
$$\cos 90^\circ = 0$$
$$\tan 90^\circ = \text{undefined}$$
Answer: \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\tan 90^\circ\) undefined
Example 2: A robotic arm moves from 0° to 90°. Calculate the sine and cosine values at the final position.
Given: \(\theta = 90^\circ\)
$$\sin 90^\circ = 1$$
$$\cos 90^\circ = 0$$
Answer: \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\)
Example 3: Determine the length of the side opposite to a 90° angle in a right triangle with hypotenuse 5 m.
Given: \(\theta = 90^\circ\), hypotenuse \(h = 5 \text{ m}\)
$$\sin 90^\circ = \frac{\text{Opposite}}{h} = 1$$
$$\text{Opposite} = h \times \sin 90^\circ = 5 \times 1 = 5 \text{ m}$$
Answer: Opposite side = 5 m
Example 4: Calculate the tangent of 89.9° as the angle approaches 90°.
Given: \(\theta = 89.9^\circ\)
$$\tan 89.9^\circ \approx 572.96$$
Answer: \(\tan 89.9^\circ \approx 572.96\)
Example 5: For a sensor calibrated at 90°, verify that the cosine value is zero.
Given: \(\theta = 90^\circ\)
$$\cos 90^\circ = 0$$
Answer: \(\cos 90^\circ = 0\)
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Create a free accountThis chapter explored various types of angles including acute, obtuse, reflex, and right angles, establishing a foundation for understanding geometric shapes. It then examined different triangles such as isosceles, equilateral, right-angled, and scalene, highlighting their unique properties. The focus shifted to trigonometric ratios, covering sine, cosine, tangent, cosecant, secant, and cotangent, essential for solving problems involving angles and lengths. Trigonometric identities were introduced next, with special attention to their proofs and the fundamental Pythagorean identities. The chapter also addressed trigonometric equations, providing methods to solve them effectively. Finally, hyperbolic functions were discussed, including sinh, cosh, cosech, tanh, and sech, broadening the understanding of functions related to exponential and trigonometric concepts in engineering mathematics.
Calculate the sine of an acute angle of \(30^\circ\) relevant to the alignment of a robotic arm in a manufacturing cell. (2 marks)
A sensor is mounted at an obtuse angle of \(120^\circ\) on a conveyor frame. Calculate the cosine of this angle. (2 marks)
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