**Pythagorean Extras**

The theorem of Pythagoras 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. The following formula holds:

sin ^{2} *θ* + cos ^{2} *θ* = 1

The expression sin ^{2} *θ* refers to the sine of the angle, squared (not the sine of the square of the angle). That is to say:

sin ^{2} *θ* = (sin *θ* ) ^{2}

The same holds true for the cosine, tangent, cosecant, secant, cotangent, and for all other similar expressions you will see in the rest of this book.

**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

**Pythagorean Extras Practice Problems**

**Use Your Calculator!**

Trigonometry is a branch of mathematics with extensive applications. You should not be shy about using a calculator to help solve problems. (Neither should you feel compelled to use a calculator if you can easily solve a problem without one.)

**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 2-2 shows a drawing of the unit circle, with the angle *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, and 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. 2-2 that the lengths of these other two sides are sin *θ* and cos *θ* . Therefore

(sin *θ* ) ^{2} + (cos *θ* ) ^{2} = 1 ^{2}

which is the same as saying that sin ^{2} *θ* + cos ^{2} *θ* = 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 of angles can be thought of as the range between, but not including, –90° and 0°.)

**Solution 2**

Figure 2-3 shows how this can be done. Draw a mirror image of Fig. 2-2, with the angle *θ* defined clockwise instead of counterclockwise. Again we have a right triangle; and this triangle, like all right triangles, must obey the Pythagorean theorem.

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