Trig Functions of Angles Right Triangle Ratios (5.2)(1)

POD Place an angle of 2π/3 in standard position. In which quadrant is the terminal side? Give the measure of a coterminal angle with a negative rotation.

The Basic Ratios We’ll start with a right triangle, the basis for all trigonometry. What does the phrase SOH-CAH-TOA mean? How does it apply to this triangle? Simplify this expression: What would that imply? a b c θ

The Basic Ratios We’ll start with a right triangle, the basis for all trigonometry. What are the reciprocal functions? (Careful, these are not the same as the inverse functions.) (Can someone explain why cosecant is the reciprocal of sine, and secant the reciprocal of cosine?) a b c θ

Match the following by connecting the lines Sin θtangent θadj/hyp Cos θsine θ hyp/adj Tan θsecant θopp/adj Sec θcosine θadj/opp Csc θcotangent θopp/hyp Cot θcosecant θhyp/opp

A couple of questions 1.Are any of these ratios ever negative in the right triangle? 2.Are any of them ever greater than 1? If so, which ones? Do any have to be greater than 1? Less than 1? Draw a chart.

Use it 1. For θ = 23°, find cos θ, sin θ, cos 2 θ, sin 2 θ and cos θ 2 on the calculators. What is cos 2 θ+ sin 2 θ?

Use it 2. For θ = π/5, find sin θ, csc θ, sin 2 θ, and sin θ 2 on calculators. What is cos 2 θ+ sin 2 θ?

Use it 3. If θ is an acute angle, and cos θ is ¾, find the values of the other five trigonometric functions of θ. (May I suggest drawing a diagram?)

Use it 4. Do you remember the special triangles? Use them to find the exact values of sine, cosine, and tangent for 30°, 45°, and 60°. (What are these angles in radian measure?) We may need to use some imagination for 90°. (We will use this information to build a unit circle. For a nice chart, look on page 375.)

Build a unit circle Using the sine and cosine values of those special triangles, let’s build a unit circle on the handout. Use colored pencils, if you wish. What do you know about the unit circle? Label the angles marked in degree and radian measure– you can use the angles in the first quadrant as reference angles for the rest of the circle, since it’s symmetric.

Build a unit circle What is the point on the positive x-axis? What is its significance? Give the coordinates for that point as it rotates in the first quadrant. Once those are done, label coordinates in the rest of the circle. How does this relate to cosine and sine?