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Pythagorean Theorem for Sine, Cosine, Secant and Tangent Help

By — McGraw-Hill Professional
Updated on Oct 25, 2011

Introduction to Pythagorean Extras—Sine, Cosine, Secant, and Tangent

The theorem of Pythagoras, which you learned earlier in this book, can be extended to cover two important facts involving the circular trigonometric functions. These are worth remembering.

Pythagorean Theorem for Sine and Cosine

The sum of the squares of the sine and cosine of an angle is always equal to 1. In mathematical terms, we can write it like this:

(sin θ ) 2 + (cos θ ) 2 = 1

When the value of a trigonometric function is squared, the exponent 2 is customarily placed after the abbreviation of the function, so the parentheses can be eliminated from the expression. Therefore, the above equation is more often written this way:

sin 2 θ + cos 2 θ = 1

Pythagorean Theorem for Secant and Tangent

The difference between the squares of the secant and tangent of an angle is always equal to either 1 or –1. The following formulas apply for all angles except θ = 90° (π/2 rad) and θ = 270° (3π/2 rad):

sec 2 θ – tan 2 θ = 1  tan 2 θ – sec 2 θ = –1

Tips and Practice Problems

Use Crutches!

Trigonometry is a branch of mathematics with extensive applications in science, engineering, architecture – even in art. You should not be shy about using a calculator and making sketches to help yourself solve problems. So what if they’re “crutches”? As long as you get the correct answer, you can use any help you need. (Just be sure you keep a fresh, spare battery around for your calculator if it’s battery powered.)

Pythagorean Theormen For Sine, Cosine, Secant, and Tangent Practice Problems

Practice 1

Use a drawing of the unit circle to help show why it is true that sin 2 θ + cos 2 θ = 1 for angles θ greater than 0° and less than 90°. (Hint: a right triangle is involved.)

Solution 1

Figure 12-10 shows a drawing of the unit circle, with θ defined counterclockwise between the x axis and a ray emanating from the origin. When the angle is greater than 0° but less than 90°, a right triangle is formed, with a segment of the ray as the hypotenuse. The length of this segment is equal to the radius of the unit circle. This radius, by definition, is 1 unit. According to the Pythagorean theorem for right triangles, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. It is easy to see from Fig. 12-9 that the lengths of these other two sides are sin θ and cos θ . Therefore:

(sin θ ) 2 + (sin θ ) 2 = 1 2

which is the same as saying that sin 2 θ + cos 2 θ = 1.

A Taste of Trigonometry USE CRUTCHES!

Fig. 12-10. Illustration for Practice 1.

Practice 2

Use another drawing of the unit circle to help show why it is true that sin 2 θ + cos 2 θ = 1 for angles θ greater than 270° and less than 360°. (Hint: this range can be considered, in directional terms, as equivalent to the range of angles greater than –90° and less than 0°.)

Solution 2

Figure 12-11 shows how this can be done. Draw a mirror image of Fig. 12-10, with the angle θ defined clockwise instead of counterclockwise. Again we have a right triangle. This triangle, like all right triangles, obeys the Pythagorean theorem.

A Taste of Trigonometry USE CRUTCHES!

Fig. 12-11. Illustration for Practice 2.

Find practice problems and solutions for these concepts at: Trigonometry Practice Test.

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