Trigonometric functions:

Concepts to Remember

• Sine, cosine, and tangent functions: Concept: The three main trigonometric functions: sine, cosine, and tangent are defined as the ratios of the sides of a right triangle opposite, adjacent, and hypotenuse to an angle, respectively. How to remember: Think of the acronym SOH-CAH-TOA.

• Pythagorean identity: Concept: The Pythagorean identity states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. How to remember: Think of the equation $$a^2 + b^2 = c^2$$ where $$a$$ and $$b$$ represent the lengths of the legs of the triangle, and $$c$$ represents the length of the hypotenuse.

• Cofunction identities: Concept: Cofunction identities relate the values of sine, cosine, and tangent of complementary angles. How to remember: Think of the word “coffee”. If you know the value of any one trigonometric function of an angle, you can use a cofunction identity to find the other trigonometric functions of the complementary angle.

• Sum and difference formulas for sine, cosine, and tangent: Concept: Sum and difference formulas express the trigonometric functions of the sum or difference of two angles in terms of functions of these angles. How to remember: You can use a mnemonic like “ALL STUDENTS TAKE CALCULUS” to help remember the sum and difference identities for sine, cosine, and tangent.

• Double-angle formulas for sine, cosine, and tangent: Concept: Double-angle formulas express the trigonometric functions of twice an angle in terms of the functions of the angle. How to remember: You can use the abbreviation “D.A.D: $$Sin^2\theta + Cos^2\theta = 1$$” to help remember the double-angle identities for sine, cosine, and tangent.

• Half-angle formulas for sine, cosine, and tangent: Concept: Half-angle formulas express the trigonometric functions of half an angle in terms of the functions of the angle. How to remember: Use mnemonic “H.A.L”: Half angle identity is just another way of expressing Pythagorean Identity. $$Sin^2\theta = \frac{1 - Cos2\theta}{2} $$

• Inverse trigonometric functions: Concept: Inverse trigonometric functions are functions that undo the trigonometric functions. How to remember: Think of arcsine, arccosine, and arctangent as the “undo” functions of sine, cosine, and tangent.

• Properties of inverse trigonometric functions: Concept: Inverse trigonometric functions have certain properties, such as their domains and ranges. How to remember: Learn the restrictions of the inverse trig functions. Just remember these restrictions: For $$y=Arcsin x$$, -1 ≤ x ≤ 1 and $-\frac{\pi}{2} ≤ y ≤ \frac{\pi}{2}$$ For $$y=Arccos x$$, -1 ≤ x ≤ 1 and 0 ≤ y ≤ π For $$y=Arctan x$$, -∞ < x < ∞ and -[\frac{\pi} {2}] < y < [\frac{\pi} {2}]

• Applications of trigonometric functions in solving real-world problems: Concept: Trigonometric functions can be used to solve various problems involving angles and triangles, such as finding the height of a building, determining the angle of elevation, and measuring distances. How to remember: Think of applications like surveying, navigation, and engineering where trigonometry is extensively used.

• Applications of Trigonometric ratios to heights and Distances: Concept: Use the right triangle relationships $$\frac{opposite}{hypotenuse} = Sin \theta$$ $$Tan \theta=\frac{Opposite}{Adjacent}$$ to find the unknown heights or distances. How to remember: Think about the basic concepts “opposite, adjacent, hypotenuse” of right triangle to understand the trigonometric ratio applications. Solving trigonometric equations: Concept: Trigonometric equations are equations involving trigonometric functions. Solving trigonometric equations involves isolating the variable and finding which values satisfy the equation. How to remember: Practice more trigonometric equation-solving problems to develop an understanding of the different techniques used.

• Using the graph of a trigonometric function to solve equations: Concept: The graph of a trigonometric function can be used to solve equations by finding the points of intersection between the graph and the x-axis or y-axis. How to remember: Visualize and analyze the graphs of trigonometric functions.