📈College Algebra Unit 10 – Further Applications of Trigonometry

Key Concepts and Definitions

• Trigonometric functions sine, cosine, and tangent represent ratios of sides in a right triangle and can be used to solve for unknown angles or side lengths
• Inverse trigonometric functions $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$ allow for finding angles when given side ratios
• Law of Sines relates the sides and angles of any triangle: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• Law of Cosines is an extension of the Pythagorean theorem for non-right triangles: $c^2 = a^2 + b^2 - 2ab \cos C$
• Oblique triangles are non-right triangles that can be solved using the Law of Sines or Law of Cosines
• Vectors are quantities with both magnitude and direction, often represented as directed line segments or using component form $(a, b)$
• Polar coordinates $(r, \theta)$ specify a point's position using distance from the origin $(r)$ and angle from the positive x-axis $(\theta)$
• Real-world applications of trigonometry include surveying, navigation, physics, and engineering problems involving triangles, vectors, and polar coordinates

Trigonometric Functions Review

• Sine $(\sin)$, cosine $(\cos)$, and tangent $(\tan)$ are the primary trigonometric functions
  • $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
  • $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
  • $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
• Reciprocal functions cosecant $(\csc)$, secant $(\sec)$, and cotangent $(\cot)$ are the reciprocals of sine, cosine, and tangent, respectively
• Trigonometric identities express relationships between trigonometric functions, such as the Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Inverse trigonometric functions, also called arc functions, are used to find angles when given side ratios
  • $\sin^{-1} (\frac{\text{opposite}}{\text{hypotenuse}}) = \theta$
  • $\cos^{-1} (\frac{\text{adjacent}}{\text{hypotenuse}}) = \theta$
  • $\tan^{-1} (\frac{\text{opposite}}{\text{adjacent}}) = \theta$
• Trigonometric functions have periodic behavior, repeating every $2\pi$ radians or $360^\circ$

Law of Sines and Its Applications

• The Law of Sines states that in any triangle, the ratio of the sine of an angle to the length of the opposite side is constant: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$
• This relationship is often written as $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
• The Law of Sines is useful for solving triangles when given:
  • Two angles and one side (AAS)
  • Two sides and an angle opposite one of them (SSA)
• In the SSA case, there may be no solution, one solution, or two solutions (the ambiguous case) depending on the given information
• The Law of Sines can be applied to solve real-world problems involving triangles, such as finding distances or heights that cannot be directly measured

Law of Cosines and Its Applications

• The Law of Cosines is an extension of the Pythagorean theorem for non-right triangles: $c^2 = a^2 + b^2 - 2ab \cos C$
• It relates the lengths of the sides of a triangle to the cosine of one of its angles
• The Law of Cosines is useful for solving triangles when given:
  • Three sides (SSS)
  • Two sides and the included angle (SAS)
• To find an angle using the Law of Cosines, use the rearranged form: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$
• The Law of Cosines can be applied to solve real-world problems involving oblique triangles, such as in navigation, surveying, and engineering

Solving Oblique Triangles

• Oblique triangles are non-right triangles that can be solved using the Law of Sines, Law of Cosines, or a combination of both
• To solve an oblique triangle, follow these steps:
  1. Identify the given information (sides and angles)
  2. Determine the appropriate law to use based on the given information
  3. Apply the law to find the missing sides or angles
  4. If necessary, use the other law to complete the solution
• When using the Law of Sines, be aware of the ambiguous case (SSA) where there may be no solution, one solution, or two solutions
• Verify your solution by ensuring that the sum of the three angles is $180^\circ$ and that the calculated sides and angles are consistent with the given information

Vectors and Vector Operations

• Vectors are quantities that have both magnitude and direction, represented by directed line segments or using component form $(a, b)$
• Vector addition follows the parallelogram law or can be performed component-wise: $(a_1, b_1) + (a_2, b_2) = (a_1 + a_2, b_1 + b_2)$
• Scalar multiplication scales a vector's magnitude without changing its direction: $k(a, b) = (ka, kb)$
• The magnitude (length) of a vector $(a, b)$ is given by $\sqrt{a^2 + b^2}$
• The dot product of two vectors $\vec{u} = (a_1, b_1)$ and $\vec{v} = (a_2, b_2)$ is a scalar value: $\vec{u} \cdot \vec{v} = a_1a_2 + b_1b_2$
• The cross product of two vectors $\vec{u}$ and $\vec{v}$ is a vector perpendicular to both: $\vec{u} \times \vec{v} = (b_1a_2 - a_1b_2)\hat{k}$
• Vectors have numerous applications in physics, engineering, and computer graphics

Polar Coordinates and Equations

• Polar coordinates $(r, \theta)$ specify a point's position using its distance from the origin $(r)$ and the angle from the positive x-axis $(\theta)$
• To convert from polar coordinates to rectangular coordinates $(x, y)$, use: $x = r \cos \theta$ and $y = r \sin \theta$
• To convert from rectangular coordinates to polar coordinates, use: $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1} (\frac{y}{x})$
• Polar equations are equations written in terms of $r$ and $\theta$, such as $r = 2\cos \theta$
• Graphs of polar equations can create interesting shapes, including circles, cardioids, and spirals
• Polar coordinates are useful in situations with circular or rotational symmetry, such as in navigation, physics, and certain branches of mathematics

Real-World Applications and Problem Solving

• Trigonometry has numerous real-world applications in fields such as surveying, navigation, physics, and engineering
• In surveying, trigonometry is used to calculate distances and angles between points, determine heights of objects, and create accurate maps
• Navigation relies on trigonometry to calculate distances, bearings, and positions using GPS or celestial navigation
• Physics and engineering problems often involve vectors, forces, and motion in two or three dimensions, which can be analyzed using trigonometric functions
• When solving real-world problems using trigonometry:
  1. Identify the given information and the quantity to be found
  2. Sketch a diagram of the situation, labeling known and unknown values
  3. Determine the appropriate trigonometric concepts or laws to apply
  4. Solve the problem using the relevant equations and techniques
  5. Interpret the solution in the context of the original problem
• Practice solving a variety of real-world problems to develop problem-solving skills and deepen your understanding of trigonometric applications