Trigonometry

Learning Objectives

• Apply SOHCAHTOA to find missing sides and angles in right triangles
• Recall sine, cosine, and tangent values for special angles (0, 30, 45, 60, 90 degrees)
• Convert between degree and radian measure
• Use the unit circle to determine trig function values in all four quadrants
• Identify the sign of trig functions in each quadrant using the ASTC rule
• Recognize and apply reference angles to evaluate trig functions for non-standard angles
• Identify amplitude, period, and phase shift from sine and cosine equations and graphs
• Apply the Pythagorean identity: sin^2(x) + cos^2(x) = 1
• Work with reciprocal trig functions (cosecant, secant, cotangent)
• Solve applied problems involving angles of elevation and depression

High-Yield Concepts

SOHCAHTOA and Right Triangle Trigonometry

SOHCAHTOA defines the three primary trig ratios in terms of a right triangle's sides relative to a given angle:

Special angle values (must be memorized):

Angles of elevation are measured upward from horizontal; angles of depression are measured downward from horizontal. In both cases, a right triangle is formed where one angle is the elevation/depression angle, and SOHCAHTOA applies to find missing heights or distances.

The Unit Circle and Radian Measure

The unit circle extends trig definitions beyond 0-90 degrees to all four quadrants. A point on the unit circle at angle theta has coordinates (cos(theta), sin(theta)). This means the x-coordinate is cosine and the y-coordinate is sine.

The ASTC rule (All Students Take Calculus) indicates which functions are positive in each quadrant:

• Quadrant I (0 to 90 degrees): All positive
• Quadrant II (90 to 180 degrees): Sine positive
• Quadrant III (180 to 270 degrees): Tangent positive
• Quadrant IV (270 to 360 degrees): Cosine positive

Radian measure: pi radians = 180 degrees. Key conversions: 90 degrees = pi/2, 45 degrees = pi/4, 60 degrees = pi/3, 30 degrees = pi/6, 180 degrees = pi, 360 degrees = 2*pi. To convert degrees to radians, multiply by pi/180. To convert radians to degrees, multiply by 180/pi.

Reference angles are the positive acute angle between the terminal side and the x-axis. To evaluate a trig function for a non-standard angle: find the reference angle (acute angle to the nearest x-axis), evaluate for the reference angle using special angle values, then apply the correct sign for the quadrant.

Graphical Properties and Identities

Sine and cosine graphs follow the form y = A sin(Bx + C) + D or y = A cos(Bx + C) + D, where:

• A = amplitude (height above and below the midline)
• Period = 2*pi / B (how long it takes to complete one full cycle)
• C/B = phase shift (horizontal translation)
• D = vertical shift (midline)

The Pythagorean identity sin^2(x) + cos^2(x) = 1 is the most important trig identity for the ACT. Rearranged forms: sin^2(x) = 1 - cos^2(x) and cos^2(x) = 1 - sin^2(x).

Reciprocal trig functions: csc(x) = 1/sin(x); sec(x) = 1/cos(x); cot(x) = 1/tan(x). The ACT occasionally tests these in simplification or substitution problems.

Study Strategy

Begin with SOHCAHTOA and right triangle applications. These are the most accessible trig questions and appear at medium difficulty on the test. Build fluency with the three ratios and their application to finding missing sides and angles.

Next, memorize the special angle table (0, 30, 45, 60, 90 degrees) for sine, cosine, and tangent. These values appear in both right triangle problems and unit circle problems, so committing them to memory eliminates computation on many questions.

Then study degree-radian conversion and the unit circle. Learn the coordinates of key unit circle points, the ASTC rule for signs, and reference angle calculation. This cluster allows you to answer questions about trig values for angles like 150, 225, and 300 degrees without memorizing a full table.

After the unit circle, study graphical properties of sine and cosine: amplitude, period, phase shift, and vertical shift. These appear in 1-2 questions per test at medium-hard difficulty.

Cover the Pythagorean identity, reciprocal trig functions, and the Law of Sines/Cosines basics last. The law of sines and law of cosines appear rarely (0-1 questions per test) and are only necessary for students targeting scores above 32.

Common Mistakes

• Setting up SOHCAHTOA with the wrong reference angle (using the other acute angle in the triangle)
• Confusing opposite and adjacent sides when the triangle is rotated or presented non-standardly
• Forgetting that angles of depression require looking at the full right triangle, including identifying where the right angle is
• Mixing up radian and degree mode on a calculator (ACT allows calculators but students must verify they are in the right mode)
• Applying ASTC signs incorrectly for third or fourth quadrant angles
• Confusing amplitude (half the peak-to-trough height) with the maximum value of the function
• Using period = B instead of period = 2*pi/B in graphical problems
• Forgetting that sec, csc, and cot are the reciprocals of cos, sin, and tan respectively (not the co-functions)
• Not checking for the reference angle before applying special angle values for non-standard angles
• Treating the Pythagorean identity as a one-direction formula; it works in either direction (solving for sin or for cos)

Exam Tips

• Sketch the right triangle before applying SOHCAHTOA, labeling the opposite and adjacent sides relative to the specific angle in question
• For unit circle problems, immediately identify which quadrant the angle falls in and apply the ASTC rule before any calculation
• For period problems, remember the period formula: period = 2*pi / B (in the equation y = sin(Bx))
• For reference angle problems: subtract from 180 for Quadrant II angles, subtract 180 for Quadrant III, subtract from 360 for Quadrant IV
• When given sin(x) or cos(x) and asked to find another trig value, draw the reference triangle and use the Pythagorean theorem
• Convert radians to degrees or degrees to radians before answering if you find one system more intuitive
• For angles of elevation and depression, clearly identify the horizontal reference line and which angle is measured from it
• Amplitude = |A| -- always take the absolute value of the coefficient before calling it the amplitude