Trigonometric Identities Help

Cosecant Of Negative Angle

The cosecant of the negative of an angle is equal to the negative (additive inverse) of the cosecant of the angle. The following formula applies for all angles except θ = 0° (0 rad) and θ = 180° ( π rad):

csc – θ = –csc θ

Secant Of Negative Angle

The secant of the negative of an angle is equal to the secant of the angle. The following formula applies for all angles except θ = 90° ( π/2 rad) and θ = 270° (3π/2 rad):

sec – θ = sec θ

Cotangent Of Negative Angle

The cotangent of the negative of an angle is equal to the negative (additive inverse) of the cotangent of the angle. The following formula applies for all angles except θ = 0° (0 rad) and θ = 180° ( π rad):

cot – θ = –cot θ

Double Angles

Sine Of Double Angle

The sine of twice any given angle is equal to twice the sine of the original angle times the cosine of the original angle:

sin 2θ = 2 sin θ cos θ

Cosine Of Double Angle

The cosine of twice any given angle can be found according to either of the following:

cos 2 θ = 1 – (2 sin ² θ )

cos 2 θ = (2 cos ² θ ) – 1

Angular Sum and Difference

Sine Of Angular Sum

The sine of the sum of two angles θ and ø can be found using this formula:

sin ( θ + ø) = (sin θ )(cos ø) + (cos θ ) (sin ø)

Cosine Of Angular Sum

The cosine of the sum of two angles θ and ø can be found using this formula:

cos ( θ + ø) = (cos θ )(cos ø) – (sin θ )(sin ø)

Sine Of Angular Difference

The sine of the difference between two angles θ and ø can be found using this formula:

sin ( θ – ø) = (sin θ )(cos ø) – (cos θ )(sin ø)

Cosine Of Angular Difference

The cosine of the difference between two angles θ and ø can be found using this formula:

cos ( θ – ø) = (cos θ )(cos ø) + (sin θ )(sin ø)

That’s enough fact-stating for now. Some of these expressions look messy, but they involve nothing more than addition, subtraction, multiplication, division, squaring, and taking the square roots of numbers you work out on a calculator.

Precedence Of Operations

When various operations and functions appear in an expression that you want to solve or simplify, there is a well-defined protocol to follow. If you have trouble comprehending the sequence in which operations should be performed, use a pencil and scratch paper to write down the numbers derived by performing functions on variables; then add, subtract, multiply, divide, or whatever, according to the following rules of precedence.

• Simplify all expressions within parentheses from the inside out
• Perform all exponential operations, proceeding from left to right
• Perform all products and quotients, proceeding from left to right
• Perform all sums and differences, proceeding from left to right

Here are a couple of examples of this process, in which the order of the numerals and operations is the same in each case, but the groupings differ.

Trigonometric Identities Practice Problems

Practice 1

Illustrate, using the unit circle model, examples of the following facts:

sin – θ = –sin θ

cos – θ = cos θ

Solution 1

See Fig. 2-4. This shows an example for an angle θ of approximately 60° (π/3 rad). Note that the angle –θ is represented by rotation to the same extent as, but in the opposite direction from, the angle θ . Generally, positive angles are represented by counterclockwise rotation from the x axis, and negative angles are represented by clockwise rotation from the x axis. The ray from the origin for – θ looks like the reflection of the ray for θ from a pane of glass that contains the x axis and is perpendicular to the page. The above identities can be inferred geometrically from this diagram. The two rays intersect the circle at points whose y values (representing sines) are negatives of each other, and whose x values (representing cosines) are the same.

Fig. 2-4. Illustration for Solution 1.

Practice 2

Simplify the expression sin (120° – θ ). Express coefficients to three decimal places.

Solution 2

Practice 3

Illustrate, using the unit circle model, examples of the following facts:

sin(180° – θ ) = sin θ

cos(180° – θ ) = –cos θ

Solution 3

Practice problems for these concepts can be found at:  Trigonometric Functions Practice Test