Understanding Sine and Cosine Graphs and Trigonometric Inverses

Sketching Graphs of Sine and Cosine with Period Changes

Ever wondered how changing the value inside sine or cosine affects its graph? When you see functions like y = sin(bx) or y = cos(bx), the value of b changes how quickly the function completes one full cycle.

The period of a trigonometric function is calculated using the formula 2π/b. For example, in y = sin(3x), the period is 2π/3, meaning the function completes a full cycle three times faster than the standard sine function. Similarly, for y = sin$π/2 × x$, the period becomes 4π/3.

To sketch these graphs, identify key points within one period. For cosine functions, these points typically include where the function equals 1, 0, -1, and back to 0. For sine functions, they're where the function equals 0, 1, 0, -1, and back to 0.

Pro Tip: When sketching trig graphs with modified periods, always calculate the new period first $2π/b$, then divide it into equal parts to find your key points. This makes the sketching process much more manageable!

Vertical Shifts and Amplitude Changes

Transforming trig graphs gets easier when you understand how different changes affect the output. When you add a constant to a function like y = cos$x$ - 2, you're creating a vertical shift that moves the entire graph down by 2 units.

To sketch these transformed functions, create a table of values for the original function first, then apply the transformation. For vertical shifts, simply add or subtract the constant from each y-value. The shape remains the same, but the position changes.

When dealing with amplitude changes like y = -5/2 cos(2x), you multiply all y-values by the coefficient (-5/2 in this case). This stretches or compresses the graph vertically, while the negative sign flips it upside down. Remember that the amplitude is the absolute value of this coefficient.

Don't forget to adjust the period too when dealing with functions like y = -5/2 cos(2x). The period changes to π $from 2π/2$, which means you'll need to recalculate your key x-values at 0, π/4, π/2, 3π/4, and π.

Remember: When sketching y = a·sin(bx) or y = a·cos(bx), the coefficient a affects the height (amplitude), while b affects the width (period) of the graph!

Phase Shifts and Inverse Trig Functions

Phase shifts occur when you have functions like y = 3cos$x + 2π/3$, which shifts the cosine graph horizontally. To find the new x-coordinates, solve for when the inside expression equals your key values $0, π/2, π, etc.$.

For example, to find where y = 3cos$x + 2π/3$ equals 3, solve x + 2π/3 = 0, giving x = -2π/3. Continue this process to find all key points, then sketch your graph using the amplitude of 3.

Inverse trigonometric functions find angles that produce specific trig values. The notation sin⁻¹$x$ means "the angle whose sine equals x." These functions have restricted domains and ranges:

• sin⁻¹$x$: Range is $-π/2, π/2$
• cos⁻¹$x$: Range is [0, π]
• tan⁻¹$x$: Range is $-π/2, π/2$

The domains and ranges of the six trig functions are essential to understand. While sine and cosine are defined for all real numbers with ranges from -1 to 1, others like tangent and secant have restrictions. Tangent and cotangent have ranges of all real numbers, but their domains exclude specific values where they're undefined.

Quick Check: When evaluating sin⁻¹$-√3/2$, remember you're finding the angle in $-π/2, π/2$ whose sine equals -√3/2, which is -π/3.