Warm-Up 1 Find the value of x.

Warm-Up 1 Find the value of x.

Trigonometric Ratios I Objectives: To discover the three main trigonometric ratios To use trig ratios to find the lengths of sides of right triangles

Summary hypotenuse side opposite Θ side adjacent Θ

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Example 3 Find the values of the six trig ratios for α and β.

Activity: Trig Table Step 5: Finally, let’s check your values with those from the calculator. For sin, cos, and tan Make sure your calculator is set to DEGREE in the MODE menu. Use one of the 3 trig keys. Get in the habit of closing the parenthesis.

Example 4 To the nearest meter, find the height of a right triangle if one acute angle measures 35° and the adjacent side measures 24 m.

Example 5 To the nearest foot, find the length of the hypotenuse of a right triangle if one of the acute angles measures 20° and the opposite side measures 410 feet.

Example 6 Use a special right triangle to find the exact values of sin(45°) and cos(45°).

Example 7 Find the area of a regular octagon with a side length of 6 inches. 22.5

Example 8 Find the value of x to the nearest tenth. x =

Find the values of 𝑤 and 𝑥

Challenge Problem Make a conjecture about how the sine and cosine of 30° angle are related.

Challenge Problem Make a conjecture about how the sine and cosine of 60° angle are related.

Challenge Problem Square ABCD in the diagram at the right has side length of 1, and midpoints of its sides are labeled P, Q, R, and S. Find the length of a side of the shaded square

Properties of Tangents Objectives: To define and use circle terminology To use properties of tangents to a circle

Tangent A tangent is a line that intersects a circle at exactly one point. The point of intersection is called the point of tangency

Tangent

Example 2 Explain why the wheels on a train are closer to being tangent to the rails than a car tire to the road.

Example 4 Draw two coplanar circles that intersect in a) two points, b) one point, c) no points and have the same center.

Common Tangents A line, ray, or segment that is tangent to two coplanar circles is called a common tangent.

Example 5 Tell how many common tangents the circles have and draw them all.

Common Tangents, II Common tangents come in two flavors: Common Internal Tangent: Intersect the segment that joins the centers of the circles Common External Tangent: Does not intersect the segment that joins the centers of the circles

Example 5, Revisited Determine whether the common tangents are internal or external.

Tangent Line Theorem In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

Example 6 The center of a circle has coordinates (1, 2). The point (3, -1) lies on this circle. Find the slope of the tangent line at (3, -1).

Example 7  

Example 8  

Example 9  

Example 10  

Congruent Tangents Theorem Tangent segments from a common external point are congruent.

Example 11  

Challenge Problem A circle has a radius of 6 inches. Two radii form a central angle of 60°. Tangent lines are drawn to the endpoints of each of the radii. How far from the center do the two tangent lines intersect? Due at end of class.

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